Method and apparatus for analyzing facial configurations and components

ABSTRACT

There is disclosed herein: (1) overlay systems which define a face, in frontal and lateral views, which are comprised of mathematically inter-related pentagons and parts of those pentagons and which establish a reference system to analyze human faces for surgical, cosmetic, and identification purposes; and (2) a method of applying the overlay systems to human faces or facial images to analyze the form and proportion of faces and components of those faces (e.g., eyes, nose, mouth, etc.) for surgical, cosmetic, and identification purposes.

This is a continuation of application Ser. No. 08/247,614, filed on May23, 1994, now U.S. Pat. No. 5,659,625, which is a file wrappercontinuation of abandoned application Ser. No. 07/894,751, filed on Jun.4, 1992, now abandoned and the above-identified applications areexpressly incorporated herein by reference.

The present invention relates to a method and apparatus for ok analyzingfacial configurations and/or facial components which can be used to: (1)plan surgical corrections or other treatments; (2) plan cosmeticapplication schemes; or (3) establish a "fingerprint" of any human facefor identification or study.

BACKGROUND

The face is the most important part of the human body for interpersonalcommunication, emotional expression, and most other forms of socialinteraction. The face is also the primary feature of the body by whichpeople recognize one another. Even newborn infants have a naturalability to recognize familiar faces.

Throughout human history, societies have placed great emphasis and valueon facial attractiveness. The significance of the aesthetics of the faceis demonstrated by the thousands of works of art and sculpture dedicatedto portraying attractive facial features. Modern research studiessuggest that human beings have a natural tendency to prefer facialconfigurations that are aesthetically pleasing. Studies have shown thatpersons whose faces are attractive are more likely to be perceived aspersonable, intelligent, and are generally perceived as possessing avariety of positive attributes. Sociological researchers have chronicledmany of the benefits enjoyed by those who are fortunate to be born witha configuration of facial features that is aesthetically pleasing. Thebenefits of having an attractive face have been shown to begin veryearly in life and to continue throughout the formative years and wellinto adulthood. It has been documented that parents tend to give moreattention and care to babies with more attractive faces. Studies alsoreveal that teachers perceive attractive children as more intelligentand popular than their less attractive peers. In comparisons of facialattractiveness, better looking high school females were found to beenjoying a higher socioeconomic status 15 years later compared to theirother classmates. Even in the military, when a class of West Pointcadets were rated for attractiveness, and later compared with their rankachieved after graduation, the more attractive cadets tended to achievea higher military rank than the less attractive cadets.

Recent sociological studies also suggest that the tendency of humans toprefer an attractive face is not learned from society or from constantexposure to the popular culture. Strong evidence suggests thatperceptions of attractiveness or beauty are generally similar in personsin different nations and even in diverse cultures.

While it is oft said that "beauty is in the eye of the beholder,"researchers have also discovered that perceptions of beauty vary littlefrom culture to culture, and that perceptions of beauty do not varysubstantially from person to person. When asked to rank photographs of20 young women in the order of attractiveness, responses from over100,000 Americans showed substantial agreement as to which photographsfeatured the most attractive faces. Furthermore, recent research hasrevealed that these perceptions begin shortly after birth. Researchershave discovered that even newly born infants show a positive reaction tophotographs of attractive faces.

Unfortunately, those born with facial deformities are likely to suffergreatly from social prejudices. Throughout history, ridicule,ostracization, and even death have been inflicted on people who sufferfrom facial deformities.

Modern medicine and surgical techniques offer hope to those born withfacial deformities or those whose faces are deformed from accident ordisease. Today, surgeons can use X-ray models, synthetic implants and avariety of other techniques to improve facial appearance. However,although modern medicine offers improved techniques and materials,neither artists, surgeons, nor sociologists are presently able tounderstand why certain faces are perceived as attractive nor are thoseskilled in the art able to offer a technique to describe the physicalparameters that, when viewed together as a human face, cause a face tobe perceived as attractive.

Many attempts throughout history have been made to develop parameters toquantify idealized facial proportions and, in essence, to quantify humanfaces. Attempts to quantify human faces initially stemmed from the Greekphilosophy that all beauty and aesthetics are based on mathematics. TheGreeks proposed that all beauty could be quantitatively determined andanalyzed, and that the essence of beauty in all things was mathematical.The Greeks showed great interest in a mathematical relationship calledthe "Golden Proportion" that is discussed in detail below. Following theRenaissance, the study of facial beauty, particularly in art, led tomultiple attempts, most notably by Leonardo Da Vinci and Albrecht Durer,to quantify facial proportions and to establish a "norm" from which allfaces could be analyzed.

In the last few centuries, the attempts to quantify facial proportionsinvolved measuring distances of and between the facial features ofpersons considered normal or attractive by the individual performing thestudy. Significant studies with large numbers of subjects were notconducted until the twentieth century. In the last one hundred years,the studies were conducted primarily by plastic surgeons andorthodontists who were continually studying faces and attempting toquantify selected facial dimensions and proportions. Although themedical and dental literature is replete with studies of the face, thereare certain inherent weaknesses in the studies.

Dentists, particularly orthodontists, are naturally interested infunction and, rather than studying attractive faces and soft tissueparameters, dental studies tend to focus on faces having normal functionand only then studying those faces for soft tissue parameters. Thus, thestudies do not disclose detailed parameters for attractive faces, ratherthe studies tend to depict functional parameters or soft tissueaesthetic parameters which are common to people with "normal function"who may or may not be attractive.

Plastic surgeons have been interested primarily in soft tissue drape andfocused their attention on measuring the parameters for ideal softtissue drape configurations. The weakness of the medical literature isthe tendency to study isolated facial soft-tissue topography as opposedto the entire face as a unit. Studies of the nasal angle, nasal frontalangle, nasal tip projection, etc., are helpful only in analyzingindividual parts of the face and give little or no information regardingan ideal or aesthetically pleasing relationship of those features to therest of the face.

The general weakness in this literature is a tendency to borrowparameters and canons for soft tissue aesthetics from the artists of theclassical and Renaissance periods. Much of the contemporary literaturein this area merely restates these classical canons adhered to by earlyartists.

Although techniques for orthodontics as well as oral surgery and plasticsurgery have become extremely sophisticated. Whether one is controllingdevelopment of the facial bones or teeth through orthodontics,restructuring facial bones through oral or reconstructive surgery, orreconfiguring facial soft tissue through plastic surgery, the technicalexpertise to perform these procedures has advanced far beyond theability to comprehensively diagnose variations from an ideal facial formbecause an apparatus or technique to analyze the face as a whole and toprovide a guide towards achieving an overall attractive appearance doesnot presently exist. Therefore, the improvements in technical skills inmedicine and dentistry in the twentieth century, particularly inorthodontics, oral surgery, and plastic surgery, heightens the need todevelop unifying parameters for an ideal facial form.

Early Greek mathematicians discovered that it is possible to describe amathematical relationship between two linear distances as follows:##STR1## such that the distance from P to Q when compared to thedistance from Q to R, has the same ratio as the distance from Q to RCompared to the distance from P to R. This relationship applies onlywhen the ratio of the distance between Q and R and the distance betweenP and Q is 1.618 18 or approximately 89/55. Greek mathematiciansrecognized the unique properties of a mathematical proportion where theratio between the greater proportion and smaller part is equal to theratio between the whole and the greater part. This proportion, typicallyindicated with the Greek letter Φ, has been called the "GoldenProportion" or the "Golden Section" and has long been used by artists,architects, and other scholars to create aesthetically pleasing works ofart and structures. Architects studying the Parthenon built on theAcropolis in Athens, Greece in the Fifth Century B.C. have noted thatthe Golden Proportion is reflected in the architectural design of thefrontal view of the Parthenon.

As used here, the greek letter Phi (Φ) represents the "goldenproportion" or its numerical value (≈89/55). Using phi, identical setsof numbers can be generated through both geometric and arithmeticprogressions. For example, setting Φ=1.618, a geometric progression canbe defined as follows: ##EQU1## the corresponding arithmetic progressionis: ##EQU2## Thus, Φ is unique. The same set of numbers is generatedboth geometrically and arithmetically and any product in the geometricprogression can be determined by adding the two preceding products inthe sequence.

These unusual properties of phi are well known. In addition, much hasbeen written about the occurrence of the phi proportion in nature. Anumber of examples of the Golden Proportion can be found in animal andplant life and the importance of the natural symmetry of the GoldenProportion has been studied in the context of biological growthmechanisms. For example, the Golden Proportion is reflected in thestructure of a number of flowers and when applied mathematically toconstruct a spiral shape, describes the appearance of certain shelltypes. The Golden Proportion was applied to human aesthetics as early asthe ancient Greeks who noted that the distances from the top of the headto the navel and from the navel to the feet were related to each other,and to the total height of the body, by the Golden Proportion Φ.

The Golden Proportion has been recognized as describing an aestheticallypleasing relationship between the sizes of the frontal upper teeth whenviewed from the front and has been applied by a number of scholars inmaking linear measurements to analyze dental aesthetics. Beyondapplication to the teeth, a number of researchers have noted theappearance of the Golden Proportion in measuring the linear distancesbetween certain points on the face. For example, in the attractive facethe width of the mouth in repose is roughly 1.618 multiplied by thewidth of the nose. In the 1946 book The Geometry of Art and Life,Professor M. Ghyka undertakes a detailed and comprehensive analysis ofthe Golden Proportion (Φ) and its application to biological systems.Professor Ghyka notes that the Φ ratio, in an "average or ideal" face,describes the vertical linear distance of the face and the distance fromthe line of the eye brows to the lower chin. The Φ ratio also is notedto describe the linear vertical distance from the lower part of the noseto the lower tip of the chin and the lower tip of the chin to themeeting line of the lips. Ghyka applied rectangles containing linesegments corresponding to the Golden Proportion to the frontal view ofthe face of an Olympic athlete. Furthermore, a practical application ofthe principles set forth by Ghyka to plastic surgery is discussed in"The Golden Proportion and Beauty" a medical journal article publishedin 1964. Doctors used the rectangular relationships noted by Ghyka torepair a severe facial deformity in a patient resulting from-untreatedfacial fractures suffered in childhood.

BRIEF DESCRIPTION OF TRE INVENTION

It has been discovered that the parameters of an attractive face and theorientation of the features that are collectively recognized as anattractive face, can be described using a collection of lines and pointsselected from a composite of pentagons whose sizes are related by theproportion and mathematical formula described above. Based on thediscovery that certain lines and points suggested by the pentagoncomposite are common to faces that are perceived as attractive, thisinvention uses selected points and lines to construct an overlay systemthat illustrates major and minor features common to beautiful faces andmay be used for several purposes, including as a guide in application ofcosmetics, as a presurgical aid for planning plastic and reconstructivesurgery, or as a standard for analyzing the face for academic study orfor quantifying the features of the face for use in an identificationsystem. In a preferred embodiment, the overlay system is applied to arecorded image of a face and variations between the face and the overlaysystem are analyzed.

This invention uses a composite of pentagons whose size is related bythe Φ proportion to describe the points and lines which are used todevelop an overlay system to describe individual soft-tissue parametersand overall configurations of an attractive face and provides an actualoverlay system for the face in repose in both the frontal and lateralview and in a smiling face in the lateral and frontal view. By applyingthe technique described here, one skilled in the art may use thepentagon complex system to analyze other facial expressions.

Moreover, this invention enables description of a large number of themajor and minor details that individually and collectively cause thefeatures of the face, and the face as a whole, to be perceived asbeautiful or attractive. By describing the elements of an attractiveface in both the lateral and frontal view, this invention enables athree-dimensional analysis of the aesthetic quality of the face.Significantly, the overlay system defined by this invention is notlimited to merely describing an ideal proportion for the distancebetween two points, but rather describes the smooth features of theface, individually and collectively, and provides a unifying model thatdescribes the individual features of the face and how they interrelateto form a face that is aesthetically pleasing.

Most applications of the overlay system to a subject's face requiresthat an image of the subject's face be obtained. A variety of techniquesmay be used to produce a facial image suitable for use with the presentinvention. The simplest is a conventional photograph or set ofphotographs showing both frontal and profile or lateral views of theface. Alternatively, and depending on the reasons for using thisinvention to analyze the face, it is possible to use a video systemconsisting of a camera and monitor. Video cameras are particularlysuitable when used with a monitor to display a fixed image of the face.Video camera compatible computer systems are available that allow animage of the face recorded with a video camera to be altered on a screenso that the visual results of a prospective surgical procedure may beanalyzed. Moreover, recent developments in computer technology allowcreation of three-dimensional models of faces that can be manipulated tobe advantageously used with the method and apparatus disclosed by thisinvention.

The use of advanced computer-image analysis systems allows one skilledin the art to use the comprehensive and detailed parameters described bythis invention as a means to create an electronic "fingerprint" that maybe used to identify persons based on their facial appearance. Onepotential application of the overlay system would be to performsophisticated measurements of the human face for use in a securitysystem. Using the overlay system described here, a computer couldanalyze the face of an individual and measure and store a large numberof parameters describing the individual's facial features based on anapplication of pentagon complexes or overlay systems to the individual'sface and a large number of measurements of the variance points. Oncethat information is stored in the computer, the computer possesses avirtual "fingerprint" of the individual's face. This process would allowthe computer to recognize individuals whose facial features have beenanalyzed and stored. Such a system would offer unique advantages whenused in a security system, as an aid to identification, in lawenforcement, or in any application where repeatable identification of anindividual's face is important. Even where a face has aged or beensurgically altered, certain parameters of the face such as the distancebetween the pupils or the distance between the ear canal and the pupilare difficult or impossible to alter surgically. Modern computertechnology could assess the deviations between two faces and determinethe probability that two faces were the same individual.

Also, a bust or moulage of the face may be made from a mold to obtain athree-dimensional facial image. As will become apparent from thefollowing description, depending on the application, one particularimage-recording device may offer advantages over another. For example,for advantageously applying cosmetics to a face using the overlay systemof this invention, a photographic print may be preferred because it issimple and inexpensive. For most cases of cosmetic or reconstructivesurgery, a collection of photographs will be satisfactory for use ofthis invention to plan and anticipate the results of a contemplatedprocedure. In other cases, computer-generated images of the face, orimages recorded with a video camera that may be altered using acomputer, may be preferred due to the enhanced capability of showing thesoft tissue of the face before and after surgery. In these cases, theoverlay system of this invention may be stored in the computer andapplied to the image of the face being generated on a computer displayscreen.

The present invention discloses apparatus and methods for theirconstruction and use, comprising a overlay system which outlines anidealized facial configuration standard and can be used to analyzeindividual facial configurations to:

(1) analyze the aesthetics of a face with respect to the overlay systemsto aid in applying cosmetics or planning surgical corrections or othertreatments to correct variations from a desired appearance (e.g.,augmentation, reduction or repositioning of facial parts or components);

(2) quantify facial proportions relative to the overlay systems orpentagon complexes; and

(3) mathematically quantify individual facial proportions, for example,for security or identification purposes.

The parameters and configuration of an attractive human face can bedepicted by selecting lines and points from a composite of pentagonstermed "pentagon complexes" whose sizes are related to each otheraccording to the proportions disclosed herein. The pentagon complexesare formed from:

(1) a primary pentagon complex which forms a basic framework for theoverlay system and from which specific lines, line segments, and pointshave been selected to construct the component lines and points of theoverlay system; and

(2) various secondary pentagon complexes, the sizes of which aremathematically related to the primary pentagon complex and from whichspecific lines, line segments, and points have been selected toconstruct the component lines and points of the overlay system.

In brief summary, to apply one preferred embodiment of the overlaysystem of this invention, the lateral view of a face in repose, thefollowing general steps are performed executed:

(1) A lateral view image of the person's face is recorded (e.g., viaphotograph, computer, etc.) ensuring that the expression of the face isproperly posed and that the head is properly positioned.

(2) Anthropometric points are established on the facial image.

(3) An overlay system containing a lateral view in repose (FIG. 1) isplaced upon the image and certain anthropomorphic points of the overlaysystem are aligned with those marked on the facial image.

(4) The lateral overlay system is sized, oriented, and adjusted to thefacial image to enable a comparison between the overlay system and theimage.

(5) The facial image is compared to the overlay system and thevariations noted or measured.

The same five basic steps (which may be performed manually or viacomputer) are followed to analyze a person's face from a frontalperspective or with a smiling expression by using a frontal view orsmiling overlay system. Thus, the overlay systems of the presentinvention, whose derivation and use is described in much greater detailbelow, may then be used to analyze a person's face to detect deviationsfrom the overlay systems. It is generally recognized that humans exhibita static facial expression known as "repose," and ten dynamic facialexpressions: happiness/joy, surprise, anger, loathing/contempt, fear,distress/sadness, interest/excitement, disgust, shame and guilt. Thefacial expressions that are perceived as the most attractive are theface in repose and the smiling face. Therefore, it is believed that theoverlay systems for these two facial expressions, as provided in FIGS. 1through 4, are the most useful analytical tools for analyzing theaesthetics of a face.

To practice this invention, FIGS. 1-4 can be used by creating atransparency directly from the figures and applying it to the image of aface. The detailed description below that describes how the individualpoints and lines that form the overlay system are derived from thevarious pentagon complexes need not be relied upon to analyze theaesthetic properties of a particular face by applying an overlay systemto an image of a subject's face. The detailed descriptions of thederivation of the overlay system of FIGS. 1-4 is provided to enable oneskilled in the art to practice this method of analysis for other facialexpressions and for the minor deviations from the overlay system inFIGS. 1-4 that have been observed in the male face.

The overlay system was derived from application of the pentagon systemand complexes of the faces of females. It is believed that the overlaysystem can be advantageously applied to the male face. Observationsuggests that the primary distinction between the ideal femaleparameters and the ideal male parameters rests primarily in a moreangular face and more prominent lower jaw. Applying one of the pentagoncomplexes described below, the repose smile complex of the frontalrepose overlay system 220, the outline of the attractive male jaw may bedescribed.

Furthermore, depending on the nature of a computer system used with thisinvention, the detailed and specific description of the lines that formthe overlay systems described here will further enable quantifying thefacial parameters.

DETAILED DESCRIPTION OF THE INVENTION

For purposes of reference and description, individual anthropometricpoints or features of the soft tissue of the face are defined as followsand as indicated in FIGS. 5, 6, 7, and 8 and may be described hereinwith the two or three letter designation indicated. For the purposes ofthis description, "anterior" indicates the direction towards the frontof the head and to the right of the printed page in FIGS. 1, 3, 5, and7. The term "posterior" indicates the direction towards the back of thehead and to the left on the printed page in FIGS. 1, 3, 5, and 7. Theterm "inferior" refers to the lower portion of the feature and to thebottom of the printed page in FIGS. 1 through 8. The term "superior"refers to the upper portion of a feature and to the top of the printedpage in FIGS. 1 through 8. As used in the context of this invention, theterm "overlay" and "overlay system" refers to a graphic representationof the features of an aesthetically pleasing face, or component partthereof, that can be compared to a subject's face for analysis. The"overlay system" does not merely describe the distances between twopoints, but provides an outline of individual facial features and theentire face that is aesthetically pleasing. In some cases, the "overlaysystem" is a tangible object that will actually physically overlay theimage of a face. In other cases, the "overlay system" may electronicallyoverlay the face such as where the "overlay system" of this inventionmay be applied to a face using a computer system. Similarly, using aspecially selected projector, the "overlay system" may be visuallyprojected onto the face to be analyzed.

Referring to the lateral or profile view in FIGS. 5 and 7, the followinganthropometric points are primary:

Primary Points

1) Pupil Point (PU)--the center of the pupil 1 seen in a frontal viewand the most anterior point of the pupil 1 seen in a profile view.

2) Tragion (T)--the notch 2 on the upper margin of the tragus 25.

3) Cheilion (CH)--the most lateral extent 3 of the outline of the lips.

4) Porion (POR)--the highest point 4 on the upper margin at thecutaneous auditory meatus.

5) Otobasion Inferius (OBI)--the point 5 of attachment of the ear lobeto the cheek.

6) Alare Posterior (AP)--the most posterior point 6 of the soft tissueoutline of the lateral cartilaginous wall of the naris.

7) Glabella (GS)--the most anterior point 7 on the soft5 tissueforehead.

8) Nasion (NS)--the most concave point 8 of the soft-tissue outline ofthe bridge of the nose.

9) Trichion (TRU)--the point 9 on the hairline in the midline of theforehead. In early childhood, identification of this landmark may bedifficult because of an irregular or indistinguishable hairline. Itcannot be determined on a balding head.

10) Pronasale (PRN)--the most anterior point 10 of the nose.

11) Menton (MES)--the most inferior point 11 of the soft-tissue outlineof the chin.

12) Pogonion (PGS)--the most anterior point 12 on the convexity of thesoft-tissue chin.

13) Inferior Palpebrion (IPR)--the most inferior point 13 of theinferior palpebral line on the right lower eyelid.

14) Subtragion (SBT)--the point 14 where the tragus joins theintertragal notch.

Again referring to FIGS. 5 and 7, the following anthropometric pointsare secondary:

Secondary Points

15) Subnasale (SBN)--the point 15 at which the columella merges with theupper cutaneous lip.

16) Vermilion Superius (VS)--the point 16 at the intersection betweenthe vermilion border of the upper lip and the cutaneous portion of theupper lip.

17) Labrale Superius (LS)--the most anterior point 17 on the convexityof the upper lip.

18) Upper Stomion (UST)--the most inferior point 18 of the anteriorportion of the upper lip.

19) Lower Stomion (LST)--the most superior point 19 on the anteriorportion of the lower lip.

20) Labrale Inferius (LI)--the most anterior point 20 on the convexityof the lower lip.

21) Vermilion Inferius (VI)--the point 21 of intersection between thevermilion border of the lower lip and the cutaneous portion of the lowerlip.

22) Gnathion (GNS)--the most everted point 22 of the soft-tissue chinbetween the Pogonion 12 and the Menton 11.

23) Gonion (GN)--the most posterior point 23 of the lower mandible.

24) Malar Point (MLR)--the most everted point 24 on the convexity of thecheek.

25) Tragus (TRG)--the soft-tissue lobe 25 between the tragion 2 and thesubtragion 14.

To practice this invention, it is necessary to properly orient the headrelative to the image-recording device. The following proceduredescribing the technique to properly orient the face before recording afrontal or lateral image, assumes the use of a conventional camera orvideo camera that may be fixed in a stable position so that the positionof the subject's head may be adjusted relative to the camera.

To record an image of the face of a subject for use with this invention,the hair should be pulled back so that the entire face and ears areexposed from the hairline to the chin bottom. The teeth should belightly held together. The facial expression should be one of reposewhere the muscles of the face are completely relaxed with the eyes openas in a naturally awake state. Proper pupil position is established bydirecting the subject to look at the reflection of their own eyes in amirror positioned at eye level. Once proper pupil position is achieved,the position of the head is carefully adjusted relative to X, Y, and Zaxes before the image is recorded. The following discussion assumes thatthe X and Z axes are horizontal axes that are parallel to the floor (orhorizontal plane) on which the subject is standing and that the Y axisis a vertical axis perpendicular to the floor on which the subject isstanding. The overlay system possesses distinct reference axes definedby the features of the face portrayed in the overlay system.

On the lateral view, the Z axis of the overlay system is a line parallelto IPR-POR and passing through a midpoint 29 which is the bisection ofPU-AP. The Y axis is a line perpendicular to the Z axis and passingthrough midpoint 29 as described above. The X axis is perpendicular toboth the Y and the Z axes passing through a midpoint 29 as describedabove. On the frontal view, the Y axis of the overlay system is a linecoincident with Γ and the X axis is perpendicual to the Y axis at thelevel of the coronal plane bisecting the line segment PU-AP,intersecting at a midpoint 30. The Z axis is perpendicular to the Y andX axes and intersects them at a midspoint 30.

As used in this description, the phrase "line of view" refers to animaginary line from the eye of the viewer to a point on the object beingrecorded. When using a conventional lens reflex or video camera, theline of view actually extends from the center of the image plane to thecenter of the image as it appears in the viewfinder of the camera.

The following procedure is followed for obtaining a properly oriented,or "true," lateral image of the face. Referring to FIG. 5, the truelateral image is obtained by adjusting the position of the head relativeto the line of view of the camera or other image-recording device suchthat a line A passing through the pupil point 1 and the alare posterior6 is perpendicular to the horizontal plane. In addition, the position ofthe head should be adjusted so that a line B passing through theinferior palpebrion 13 and the porion 4 is parallel to the horizontalplane. When lines A and B are oriented as described, the lateral view ofthe face is thus initially oriented relative to the Y and Z axes.

To achieve proper orientation of the lateral image view of the facerelative to the Y axis, the position of the head is adjusted so that amidsagittal plane of the head is perpendicular to the horizontal plane.Proper rotation about the Y and Z axis is achieved by verifying that theimage plane of the camera is parallel to the midsagittal plane. Theheight of the camera should be adjusted so that the line of view isperpendicular to the midsagittal plane at a midpoint 29 that bisects thesegment of the line A between the pupil point 1 and the alare posterior6 (line segment PU-AP). This establishes the proper orientation relativeto the X axis.

Although a rough approximation of the proper head position may beachieved by extremely careful visual inspection, it is preferable inpractice to achieve a more precise orientation of the face by actuallymarking important lines and points on the face as an aid to visualizingproper head position or by projecting horizontal and vertical lines,such as lines A and B in FIG. 5, onto the face and adjusting the headposition so that the points indicated above fall on the projected lines.

It is also possible to project the overlay system in FIGS. 1 through 4directly onto the subject's face using specially selected projectionequipment. To project the overlay, a lens system is needed that will notenlarge or reduce the size of the overlay as it is projected throughspace.

Once proper orientation in the X, Y, and Z axes is achieved, the facialexpression of absolute repose is posed and the image recorded. Ifanother facial expression is also to be analyzed, that expression isposed without altering the head position and the image recorded. Afterall desired images are recorded, each of the primary and secondaryanthropometric points of the soft tissue of the face are identified anddesignated on the recorded image of the face in absolute repose. If aphotograph of the face has been obtained, one may simply place marks onthe image corresponding to the primary and secondary points as describedherein.

Once the points are marked on the recorded image of the subject's face,the overlay system that is the subject of this invention can be appliedto that image. Preferably a transparency containing the overlay systemis placed onto a photograph of the face on which the primary andsecondary anthropometric points have been marked.

The proper size and orientation of the overlay system is determined bycomparing the location of certain points, and the length and location ofcertain line segments, on the facial image and on the overlay system.The technique described here for positioning the head with the face inrepose is also used to position the head to the smiling face and otherfacial expressions. To record an image of facial expressions other thanrepose, the head is first positioned with the face in repose, then thedesired facial expression is posed just prior to the image beingrecorded.

To create the line segments described here, a straight line of finitedistance is drawn between, and terminates on, two points. Thus, a linesegment from a point A to a point B, would be designated as AB herein,would contain points A and B as endpoints and would have a length equalto the minimum distance between points A and B.

Referring to FIG. 5, the pupil point 1 of the overlay system should be"coincident," meaning placed directly on top of so as to partially ortotally cover the pupil point 1 of the subject's facial image. When thepupil point 1 of the overlay system and the facial image are aligned,the size of the overlay system relative to the facial image must beadjusted. To do so, the size of the entire lateral overlay system isreduced or enlarged so that the length of the line segment from thepupil point 1 to the cheilion 3 PU-CH in the overlay system is equal tothe length of the line segment PU-CH in the facial image. The propersize of a lateral overlay system is verified by comparing, between theimage of the face and the lateral overlay system, the length of the linesegment from: (1) the pupil point 1 to the alare posterior 6 (linesegment PU-AP), (2) the pupil point 1 to the menton 11 (line segmentPU-MES), (3) the pupil point 1 to the porion 4 (line segment PU-POR),and (4) the alare posterior 6 to the porion 4 (line segment AP-POR).

Once the length of the PU-CH line in the overlay system is correct, andverified with the four other line segments, the overlay system isproperly sized in the Y and Z axis.

To establish the proper orientation of the lateral overlay system in theY-Z plane, the overlay system transparency is rotated around the pupilpoint 1, maintaining coincidence of the pupil points 1 on the overlaysystem and pupil point 1 on the facial image. The final orientation ofthe overlay system is set by visually examining the relative positionsof lines and points on the facial image and the overlay system. Insetting the final orientation of the overlay system relative to theface, primary consideration is given to placing line segment PU-POR ofthe overlay system coincident to or parallel with the PU-POR line of thefacial image or, alternatively, tangent to the POR point 4 of the facialimage. Secondary consideration is given to adjusting the overlay systemsuch that the PU-AP line coincides with the PU-AP line of the facialimage. Proper position of the lateral overlay system in the Y-Z plane isverified by insuring approximate coincidence between the tragus 25 andthe otobasion inferius 5 in the lateral overlay system and the facialimage.

To finally orient the overlay system on the Z axis, the overlay systemis moved along the Z axis to achieve a best fit between the primaryanthropometric points of the overlay system and the corresponding pointsof the face. In determining a best fit, aesthetic judgment may guide thefine adjustments to the positioning of the overlay system relative tothe facial image. For example, if the outline of the chin and jaw aloneare being analyzed, the remaining points on the facial image would begiven priority in fixing the location of the overlay system so that thesubject's chin and jaw could be compared to the position of the chin andjaw in the overlay system. In most cases, less priority is given toachieving coincidence of the ear features because it has been discoveredthat the location of the ear may vary in individuals who otherwisepossess highly attractive facial features.

Therefore, the analyst may make numerous fine adjustments by changingthe orientation of the overlay system to the study of the points on theprofile of the face that are of interest.

Generally, the following points are significant in creating theaesthetic appearance of the face: Pupil Point 1, Cheilion (CH) 3, AlarePosterior 6, Glabella Point (GS) 7, Nasion Point (NS) 8, Trichion Point(TR) 9, Pronasale Point (PRN) 10, Menton (MES) 11, Pogonion (PGS) 12,Subnasale (SBN) 15, Vermillion Superius (VS) 16, Labrale Superius (LS)17, Upper Stomion (UST) 18, Lower Stomion (LST) 19, Labrale Inferius(LI) 20, Vermillion Inferius (VI) 21, and Gnathion (GNS) 22.

These last fine adjustments now establish the position of the overlay onthe lateral repose face image and this in turn establishes the true Yand Z axes of the subject face. The amount-of variance between thesegment of the B POR-IPR on the facial image and a line segment POR-IPRon the overlay system is noted. If any variance exists, the POR-IPRsegment of the overlay system should be transferred to the face tocreate a hypothetical POR-IPR segment for later application to theorientation technique for the frontal view.

The principles that are used to position the head to properly record theimage of the frontal view of the face are similar to those describedabove for the lateral view. In practice, when the lateral view isproperly fixed initially, proper orientation of the frontal view is moreprecisely obtained. Therefore, it is preferable to fix the lateralorientation of the head and face, record the lateral image, and thenre-position the image-recording device and make minor adjustments to thehead position to obtain the frontal images.

Referring to FIG. 7, proper rotation of the head for the frontal viewrelative to the X axis is achieved by adjusting the position of the headuntil the line B in the lateral view that contains the inferiorpalpebrioh (IPR) 13 and the porion (POR) 4 is parallel to the horizontalin the true lateral view. To avoid introducing an error in the frontalview relative to the position in the X axis that may be caused by amisalignment of the inferior palpebrion (IPR) 13 and the porion (POR) 4in the subject's face, the amount of variance in the position of theline segment POR-IPR of the subject's face and the POR-IPR line segmentof the overlay system, which was noted in setting the lateral view, isexamined. If the POR-IPR line in the lateral view of the subject's facevaried from the POR-IPR line segment of the lateral overlay system, thePOR-IPR line segment (or sufficient reference points to allow thesegment to be visualized) from the lateral overlay system wastransferred to the facial image and the subject's head oriented so thatthis transferred line is parallel to the horizontal. When the IPR-PORline segment is properly aligned with the horizontal, the frontal viewis properly oriented relative to the Z axis.

To achieve proper orientation relative to the X axis, the midsagittalplane of the head should be perpendicular with the horizontal. Properrotation relative to the Y axis is obtained by verifying that thefrontal plane of the face is parallel to the image plane of the imagerecording device and that the line of view is contained in themidsagittal plane. The orientation of the frontal view is correct whenthe axes of the frontal view and-the axes of the lateral view arecoincident. Also, when the head is properly oriented, the X axis of thefrontal view is perpendicular to the Y and Z axes of the lateral view.

Finally, the height of the image recording device is adjusted so thatthe line of view is perpendicular to the frontal plane, coincident withthe midsagittal plane, and coincident with a coronal plane at a verticallevel such that the coronal plane bisects the line segment PU-AP. Thisestablishes proper orientation of the image recording device. As withthe lateral view, the entire face should be exposed from hairline tochin bottom including the ear, the facial expression of absolute reposeis confirmed, and the image recorded. If another facial expression isalso to be analyzed, that expression is posed without altering the headposition and the image recorded.

After all desired images are recorded, each of the primary and secondaryanthropometric points of the soft-tissue of the face are identified anddesignated on the recorded image of the face. using the recorded imageof the frontal view of the face, the anthropometric points areestablished as indicated below. The following points are designated asprimary points and are consistent with the descriptions given above forthe lateral view: right pupil point 1, left pupil point 1', rightcheilion 3, left cheilion 3', right alare posterior 6, and left alareposterior 6'.

The following points are designated as secondary points of the frontalview and are consistent with the descriptions given above as used withthe lateral view: trichion 9, menton 11, right malar point 24, leftmalar point 24' right gonion point 23, and left gonion point 23'.

Referring to FIG. 7, to properly apply the overlay system to the frontalview of the facial image, a midline is established by constructing thefollowing line segments. Referring to FIG. 2, a first line segment 1-1'is drawn from the right pupil point 1 to the left pupil point 1'. Asecond line segment 3-3' is drawn from the right cheilion 3 to the leftcheilion 3'. A third line segment 6-6' is drawn from the right alareposterior 6 to the left alare posterior 6'. The mid-points 26, 27, and28 of the three line segments 1-1', 3-3', 6-6' are measured and markedand a straight vertical line r that attempts as much as possible tointersect the three mid-points 26, 27, and 28 respectively are drawn.

As with the lateral repose overlay system of FIG. 1, the frontal reposeoverlay system of FIG. 2 is best used by creating a transparency of FIG.2 and overlaying the transparency onto a properly oriented photograph ofthe frontal view of the face.

To orient the frontal overlay complex relative to the Y axis, the line Γof the frontal overlay system is placed coincident to the line Γ of thefacial image.

The proper size of the frontal overlay system is determined by enlargingor reducing the overlay system such that (1) the right pupil point 1 andleft pupil point 1' of the overlay system lie on line segment 1-1' ofthe facial image, and (2) the right cheilion 3 and left cheilion 3' lieon line segment 3-3' of the frontal overlay system.

The correct vertical size for the frontal overlay complex is verified byevaluating the correlation of the length of the line segment between thetrichion 9 and the menton 11 (TRU-MES) in the facial image and thefrontal overlay system. The horizontal size is verified by evaluatingthe length of the line segment between the right malar point 24 and theleft malar point 24' and the right gonion point 23 and the left gonionpoint 23', respectively, as well as facial outline.

A final adjustment of the orientation of the frontal overlay complex isaccomplished by visually performing a best fit of the primaryanthropometric points while maintaining approximate coincidence of lineΓ of the facial image and line Γ of the frontal overlay system. Inmaking the final adjustment, the secondary points, trichion 9, mention11, right gonion 23, and left gonion 23', and right malar 24 and leftmalar 24' points are considered as are the remaining anthropometricpoints of the frontal view. Aesthetic judgment and the particular use towhich the invention is applied should be considered when applying thefrontal overlay system as in the application of the lateral reposeoverlay system.

Moreover, due to the wide variety of potential uses to which the overlaysystem may be advantageously applied, the user of the overlay system maychoose to analyze the face or an individual feature in a multitude ofways. The technique described above instructs on how to achieve theproper orientation of the head so that the overlay system disclosed herecan be consistently and advantageously applied. Once the overlay systemis properly sized to fit a properly oriented facial image, the user ofthis invention may then apply the knowledge thus obtained in severalways. For example, if the overlay system suggests that the face would beaesthetically enhanced by de-emphasis of a particular feature, the usermay apply cosmetics to achieve the desired effect. The surgeon may usethe same information to plan a procedure and then check a hypotheticalpost-operative face against the overlay system to determine whether theresult of the surgery will be desirable in terms of the overall harmonyof the facial features. Alternatively, the researcher may use theoverlay system to qualitatively or quantitatively study the facialfeatures for medical or sociological studies, and/or security oridentification purposes.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is an overlay system of the lateral view of a face in repose.

FIG. 2 is an overlay system of the lateral view of a smiling face.

FIG. 3 is an overlay system of the frontal view of a face in repose.

FIG. 4 is an overlay system of the frontal view of a smiling face.

FIG. 5 is a lateral view of a face in repose where the anthropometricpoints, lines, and line segments used in positioning the head andapplying the overlay system have been marked and the points and linesegments needed to size and orient the overlay system relative to thefacial image are indicated.

FIG. 6 is a lateral view of a smiling face marked with anthropometricpoints.

FIG. 7 is a frontal view of a face in repose where the anthropometricpoints, lines, and line segments used in positioning the head andapplying the overlay system have been marked and the points and linesegments needed to size and orient the overlay system relative to thefacial image are indicated.

FIG. 8 is a frontal view of a smiling face marked with anthropometricpoints.

FIG. 9A shows an overlay system of the present invention including afrontal view of a repose facial configuration.

FIG. 9B shows an overlay system of the present invention including alateral view of a repose facial configuration.

FIG. 10A shows an overlay system of the present invention including afrontal view of a smiling facial configuration.

FIG. 10B shows an overlay system of the present invention including alateral view of a smiling facial configuration.

FIG. 11 shows a pentagon complex of the present invention.

FIG. 12A shows a base pentagon system of the present invention.

FIG. 12B shows a regular base pentagon.

FIG. 12C shows the vertices of the regular base pentagon of FIG. 12B.

FIG. 12D shows the center of the regular base pentagon of FIG. 12B.

FIG. 12E shows the center of the regular base pentagon of FIG. 12B.

FIG. 12F shows the primary reference axis of the regular base pentagonof FIG. 12B.

FIG. 12G shows the primary radials of vertex "A" of the regular basepentagon of FIG. 12B.

FIG. 12H shows the primary radials of vertex "A" and vertex "B" andtheir intersects of the regular base pentagon of FIG. 12B.

FIG. 13A shows an inverse pentagon system of the present invention.

FIG. 13B shows an inverse regular pentagon.

FIG. 13C shows the vertices of the inverse regular pentagon of FIG. 13B.

FIG. 13D shows the center of the inverse regular pentagon of FIG. 13B.

FIG. 13E shows the center of the inverse regular pentagon of FIG. 13B.

FIG. 13F shows the primary reference axis of the inverse regularpentagon of FIG. 13B.

FIG. 13G shows the primary radials of vertex "F" of the inverse regularpentagon of FIG. 13B.

FIG. 13H shows the primary radials of vertex "F" and vertex 20 "G" andtheir intersects of the inverse regular pentagon of FIG. 13B.

FIG. 14A shows concentric pentagons each of which is related in size bya factor of 1/Φ times the next larger pentagon.

FIG. 14B shows the size relation between a base pentagon and pentagonssized:

(1) Φ/2 times the base pentagon; and

(2) 2/Φtimes the base pentagon.

FIG. 14C shows the size relation between a base pentagon and pentagonssized:

(1) (Φ)^(1/3) times the base pentagon;

(2) (Φ)^(2/3) times the base pentagon; and

(3) (Φ)^(3/3) or Φ times the base pentagon.

FIG. 15 shows a matrix pentagon complex of the frontal repose overlaysystem.

FIG. 16 shows a pentagon complex size n=0, Z=1.

FIG. 17 shows a pentagon complex size n=1, Z=1.

FIG. 18 shows a pentagon complex size n=2, Z=1.

FIG. 19 shows a pentagon complex size n=2, Z=Φ/2.

FIG. 20 shows a pentagon complex size n=2, Z=2/Φ.

FIG. 21 shows a pentagon complex size n=2, Z=2/Φ×Φ^(1/3).

FIG. 22 shows a pentagon complex size n=3, Z=1.

FIG. 23 shows a pentagon complex size n=3, Z=Φ^(1/3).

FIG. 24 shows a pentagon complex size n=3, Z=Φ^(2/3).

FIG. 25 shows a pentagon complex size n=4, Z=1.

FIG. 26 shows a pentagon complex size n=5, Z=1.

FIG. 27 shows a pentagon complex size n=5, Z=Φ/2.

FIG. 28 shows a pentagon complex size n=6, Z=1.

FIG. 29 shows a pentagon complex size n=6, Z=1.

FIG. 30 is a flow diagram of a method for recording and analyzing animage of a human face according to the invention.

FIG. 31 is a flow diagram of a method for creating a system to analyzethe human face according to the invention.

FIG. 32 is a flow diagram of a method to correct variations from adesired appearance of the human face according to the invention.

FIG. 33 is a flow diagram of a method to construct a two-dimensionalgeometric form to describe the form of the human face according to theinvention.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

One embodiment of this invention comprises overlay systems derived frommathematically inter-related pentagons and specific components of thosepentagons. All pentagons referred to herein are regular pentagons,defined as having sides of equal length and having equivalent internalangles. These pentagons can be combined to form a composite such thatselected points and lines of the composite define the parameters of anaesthetically attractive face. FIGS. 9A, 9B, 10A, and 10B show preferredoverlay systems 110a, 110b, 110c , and 110d comprising frontal (FIG. 9A)and lateral (FIG. 9B) views of a repose facial configuration and frontal(FIG. 10A) and lateral (FIG. 10B) views of a smiling facialconfiguration.

Turning to the overlay systems 110a-d of the present invention; FIGS. 9Aand 9B show two-dimensional representations of frontal and lateral viewsof a facial configuration standard in repose. It will be apparent to oneskilled in the art that these two-dimensional representations could beprojected into a three-dimensional system to form a three-dimensionaloverlay structure. Therefore, the present disclosure will primarilydescribe the two-dimensional overlay systems 110a-d.

Turning to the structure of the present invention, the overlay systems110a-d comprise multiple pentagon complexes 112 (FIG. 11). Each of themany pentagon complexes making up the overlay systems comprise a basepentagon system 114 (FIG. 12A) and an inverse-pentagon system 116 (FIG.13A). The base pentagon system 114 (FIG. 12A) and the inverse-pentagonsystem 116 (FIG. 13A) each comprise a regular base pentagon 118 (FIGS.12B and 13B) and specific components of those pentagons as describedbelow.

Each base pentagon system 114 (FIG. 12A) comprises: a base pentagon 118(FIG. 12B); five vertices 120 (designated A-E and which may beidentified by, e.g., "V(A)" to describe vertex A) (FIGS. 12A, 12C-12H);a center point 122 (FIGS. 12D and 12E); a primary reference axis 124(FIG. 12F); radials 126 (FIGS. 12G and 12H); and intersect points 128(FIG. 12H).

The center point 122 (FIG. 12D and 12E) of a pentagon system may bedefined as: (1) a center of a circle 129 tangent to the vertices 120 ofthe pentagon 118 and circumscribing the pentagon 118; or (2) anintersect of the radials 126 extending from the vertices 120 of apentagon and bisecting a side 127 of the pentagon 118 opposite thevertices 120. The primary reference axis 124, as shown in FIG. 12F,comprises a line bisecting the "A" vertex 120 (e.g. "V(A)") of the basepentagon 118, passing through the center 122 of the base pentagon 118and bisecting the side 127 of the pentagon 118 opposite the "A" vertex.

Radials 126 (FIG. 12G and 12H) of the pentagon system include primaryradials 130, secondary radials (not shown), and tertiary radials (notshown). Primary radials 130 are lines extending from vertices 120 of apentagon every eighteen degrees such that two radials 126 coincide withtwo sides 127 of the pentagon 118 and one radial 126 is parallel to orcoincident with the primary reference axis 124 (see FIGS. 12A, 12G, and12H). Secondary radials (not shown) are lines extending from vertices120 of a pentagon 118 which are not primary radials 130. Tertiaryradials (not shown) are lines which extend from intersect points 128(described below).

Each vertex has twenty primary radials 130 which are numbered ineighteen degree increments from zero degrees to three hundred sixtydegrees starting with a radial 126 which is parallel to or coincidentwith the primary reference axis 124 of the base pentagon 118 and whichextends in the same direction as a segment of the primary reference axis124 which extends from the "A" vertex 120 external to the base pentagon118 (see FIGS. 12G and 12H). Radials 126 are identified by a designatingthe vertex and degree from which they extend. For example, "V(X) R(120)"describes a radial 126 extending from a particular vertex 120 "X" of apentagon system 112 at one hundred and twenty degrees relative to aradial which is parallel to or coincident with the primary referenceaxis 124 of the pentagon system 112. In further example, the primaryreference axis 124 and any radials 126 parallel to or coincident withsaid axis 124 are designated as "V(X) R(0/360)" where X is thecorresponding vertex 120 and 0/360° is the degree designation.

Intersect points 128 include primary intersect points 132, secondaryintersect points (not shown), and tertiary intersect points (not shown).Primary intersect points 132 (FIG. 12H) are points 128 at which aprimary radial 130 intersects another primary radial 130. Secondaryintersect points (not shown) are points 128 at which a secondary radial(not shown) intersects either another secondary radial (not shown) or aprimary radial 130. Tertiary intersect points (not shown) are points 128at which a tertiary radial (not shown) intersects either anothertertiary radial (not shown) or a secondary radial (not shown) or aprimary radial 130. Intersect points 128 are identified by designatingthe two radials 126 which intersect to form the intersect point 128. Forexample, "V(X) R(234) I V(Y) R(342)" describes an intersect point 128defined by the intersection of a radial 126 extending from vertex "X" attwo hundred thirty four degrees and a radial 126 extending from vertex"Y" at three hundred forty two degrees.

The inverse pentagon system 116 comprises all the features of the basepentagon system 114 described above. However, the inverse pentagonsystem 116 is rotated one-hundred eighty degrees with respect to thebase pentagon system 114 (compare FIGS. 12A-12H and FIGS. 13A-13H). Theinverse pentagon system 116 comprises a base pentagon 118 (FIG. 13B),five vertices 120 (labeled F-J) (FIG. 13C), a center point 122 (FIGS.13D and 13E), a primary reference axis 124 (FIG. 13F), radials 126(FIGS. 13G and 13H), and intersect points 128 (FIG. 13H). The centerpoint 122 and the primary reference axis 124 of the inverse pentagonsystem 116 coincide with the center point 122 and the primary referenceaxis 124 of the base pentagon system 116 (see FIG. 11).

As mentioned above, the pentagon complex 112 (FIG. 11) comprises a basepentagon system 112 and an inverse pentagon system 116. Thus, thepentagon complex 112 comprises an aggregate of the features of the basepentagon 114 and inverse pentagon 116 systems described above.

A conglomeration of various pentagon complexes 112 form the basis of theoverlay systems 110a-d of the present invention. Fragments of thevarious pentagon complexes 112 make up the overlay systems 110a-d. Indescribing these fragments, it is necessary to identify vertices 120,radials 126, and intersects 132. As described above, vertices 120 areidentified by designations such as "V(X)" to identify vertex X andradials 126 are identified by designations such as "V(X) R(180)" toidentify a radial extending from vertex X at an angle of one hundredeighty degrees. An intersect point 128 is identified by designating theradials 126 which intersect to make the point 128. The intersectingradials 126 which define a particular 20 intersect point 128 are notnecessarily radials 126 from the same pentagon complex 112. Thus, in adescription of an intersect point 128, the pentagon complexes 112 fromwhich the radials 126 defining the intersect point 128 extend should beclear. However, generally, when a particular pentagon complex 112 isbeing described the designation of the particular complex 112 from whichthe intersecting radials 126 extend is not written in every intersectpoint 128 description. Rather, when describing an intersect point 128 ofa particular complex 112 it is assumed that the vertices 120 and radials126 listed are of the complex 112 at issue unless some other complex 112is designated. For example, when describing a particular pentagoncomplex "Z" a proper designation of an intersect point 128 may be: "V(A)R(180) I V(B) R(270)" (see 131 of FIG. 12H) meaning a point 128 at whicha radial 126 extending from vertex A of complex Z at one hundred eightydegrees intersects a radial 126 extending from vertex B of complex Z attwo hundred seventy degrees. However, if the description of an intersectpoint 128 requires using a radial 126 from a complex 112 other than thecomplex 112 at issue (i.e. a radial from a complex 112 other than "Z"),then the complex 112 must be designated in the intersect point 128description. For example, "complex Z V(C) R(18) I complex Y V(H) R(90)"indicates the point at which a radial 126 extending from vertex C ofcomplex Z at eighteen degrees intersects a radial 126 extending fromvertex H of complex Y at ninety degrees. Thus, when an intersect point128 is described without any complex 112 designations it is assumed thatthe vertices 120 and radials 126 listed are components of the complex112 being described.

The overlay systems 110a-d of the present invention comprise multiplepentagon complexes 112 like the pentagon complex 112 described above.The multiple pentagon complexes 112 are all related in size and positionto each other and to a matrix or reference pentagon complex 134.

The sizes of the pentagon complexes 12 are related by the followingformula:

    PC  subject!=PC  reference!× 1/Φ!.sup.n ×Z

where

PC subject!=the size of the target pentagon complex

PC reference!=the size of the reference pentagon complex

Φ=(1.618 . . .

n=0-6

Z=1; 2/Φ; Φ/2; (Φ)^(1/3) ; (Φ)^(2/3) ; or a combination of thesemultiplied together.

FIGS. 14A-14C show pentagons of different sizes as described accordingto this formula. FIG. 14A shows pentagons the sizes of which are definedby the above formula with n=0-6. Thus, a reference pentagon 134 is shownas the largest pentagon (and, therefore, has a size defined by n=0), thesize of the next smaller pentagon 136 is defined by n=1, the size of thenext smaller pentagon 138 is defined by n=2, and so on.

In addition, pentagons shown in FIG. 14B are pentagons the sizes ofwhich are defined by the above formula where they have the same valuefor n but differing values for Z in Z=2/Φ and Φ/2. Thus, a referencepentagon 134 is shown as a mid-sized pentagon (Z=1), the smallerpentagon 140 is defined by Z=Φ/2, and the larger pentagon 142 is definedby Z=2/Φ.

Furthermore, FIG. 14C shows pentagons the sizes of which are alsodefined by the above formula (with constant "n" values), butZ=(Φ)^(1/3), (Φ)^(2/3), and (Φ)^(3/3) (i.e., (Φ)^(1/3) times (Φ)^(2/3)).Thus, a reference pentagon 134 is shown as the smallest pentagon, thesize of the next larger pentagon 144 is defined by Z=(Φ)^(1/3), the sizeof the next larger pentagon 146 is defined by Z=(Φ)^(2/3), and the sizeof the next larger pentagon 148 is defined by Z=(Φ)^(3/3) or Φ.

Thus, all the pentagon complexes in a particular overlay system arerelated by a factor of phi ("Φ"), the golden proportion.

As mentioned above, each of the overlay systems 110a-d of the presentinvention comprise a matrix (or reference) pentagon complex 134 andfragments of pentagon complexes 112 of varying sizes and positions. Thepositions of pentagon complexes 112 in a particular overlay system110a-d may be described with respect to the matrix pentagon complex .134of that overlay system 110a-d. For example, the position of a particularvertex 120 or intersect point 128 or radial 130 of a pentagon complex112 may be described as coinciding with an intersect point 128 or radial130 of the matrix pentagon complex 134 (see below for actual examples).In addition, any tilt of the pentagon complex 134 with respect to thematrix pentagon complex 134 may be described by indicating a coincidingor parallel radials 126 of the pentagon complex 112 in question and thematrix pentagon complex 134.

Because the pentagon complexes 112 which make up the overlay systems110a-d are all related, some lines of a particular system 110a-d may bedescribable as fragments of more than one particular pentagon complex112. Therefore, in the detailed descriptions below some lines are or maybe described more than once. In addition, the close relation of thevarious complexes 112 makes it possible to describe subtle variations-incertain features represented by the overlay systems 110a-d. Therefore,in the detailed description below some lines may describe suchvariations. Furthermore, for clarity and due to the minute detail of theoverlay systems 110a-d some of the line fragments and variationsdescribed may not be indicated on the figures provided. However, thesefragments are easily drawn by one skilled in the art from the detaileddescription given below.

As mentioned above, a detailed description of the individual points andlines that form the overlay systems will now be provided to enable oneskilled in the art to practice the present invention.

I. Frontal Repose Overlay System

A overlay system 110a comprising a frontal view of a face in repose(FIG. 9A) will be described-by first describing a matrix complex 134 andthen describing other complexes making up the particular overlay system.In describing the other pentagon complexes 112, the size of the pentagoncomplexes 112 (starting with those which are the smallest) will bedescribed first, then their positions relative to complex 134, and thenpoints, lines, and/or areas defined by the pentagon complexes 112 whichoutline the overlay system 110a in interest. After the smallest pentagoncomplexes 112 are described, additional pentagon complexes 112 will beaddressed in order of increased sizes. There are seven different sizesof pentagon complexes 112 involved in the overlay system 110a comprisinga frontal view of a face in repose.

A. Matrix Complex

The matrix complex 134 is, by definition, of the size n=0 and Z=1 (SeeFIGS. 15 and 16). Features of the overlay system 110a are defined byfragments of this complex 134 comprising: (1) a portion of a radialextending from V(A) at 126° and extending between intersect V(A) R(126)I V(F) R(270) and intersect V(A) R(126) I V(F) R(180) 101; (2) a portionof a radial extending from V(A) at 234° and extending between intersectV(A) R(234) I V(J) R(90) and intersect V(A) R(234) I V(J) R(180) 102;(3) a portion of a radial extending from V(F) at 180° and extendingbetween intersect V(F) R(180) I V(B) R(306) and intersect V(F) R(180) IV(B) R(216) 103; (4) a portion of a radial extending from V(J) at 180°and extending between intersect V(J) R(180) I V(E) R(54) and intersectV(J) R(180) I V(E) R(144) 104; (5) a portion of a radial extending fromV(F) at 252° and extending between intersect V(F) R(252) I V(A) R(126)and intersect V(F) R(252) I V(A) R(180) 105; (6) a portion of a radialextending from V(J) at 108° and extending between intersect V(J) R(108)I V(A) R(234) and intersect V(J) R(108) I V(A) R(180) 106; (7) a portionof a radial extending from V(F) at 198° and extending between intersectV(F) R(198) I V(A) R(126) and intersect V(F) R(198) I V(A) R(144) 107;(8) a portion of a radial extending from V(J) at 162° and extendingbetween intersect V(J) R(162) I V(A) R(234) and intersect V(J) R(162) IV(A) R(216) 108; (9) a portion of a radial extending from V(A) at 144°and extending between intersect V(A) R(144) I V(F) R(198) and intersectV(A) R(144) I V(F) R(180) 109; (10) a portion of a radial extending fromV(A) at 216° and extending between intersect V(A) R(216) I V(J) R(162)and intersect V(A) R(216) I V(J) R(180) 110; (11) a portion of a radialextending from V(J) at 90° and extending between intersect V(J) R(90) IV(A) R(234) and intersect V(J) R(90) I V(A) R(126) 111; (12) a portionof a radial extending from V(A) at 162° and extending between intersectV(A) R(162) I V(F) R(216) and intersect V(A) R(162) I V(F) R(252); (13)a portion of a radial extending from V(A) at 198° and extending betweenintersect V(A) R(198) I V(J) R(108) and intersect V(A) R(198) I V(J)R(144); and (14) a portion of a radial extending from V(B) at 270° andextending between intersect V(B) R(270) I V(A) R(162) and intersect V(B)R(270) I V(A) R(198).

B. Pentagon Complexes with n=6

There are only two pentagon complexes 112 of the size n=6 (See FIG. 29).They are iris pentagon complexes 150 and define right and left irises onthe frontal repose overlay system 110a. These pentagon complexes have Zvalues of 1.

1. Right Iris Complex

The iris pentagon complex 150 which defines the right iris on theoverlay system 110a is positioned on the matrix complex with V(E) of theiris complex 150 coinciding with intersect matrix complex V(E) R(108) Imatrix complex V(F) R(234) and iris complex V(E) R(108) coinciding withmatrix complex V(E) R(108). Features of the overlay system 110a aredefined by fragments of this pentagon complex comprising: (1) a circlecircumscribing the pentagon complex and intersecting the vertices (e.g.V(A), V(B), V(C), . . . etc.); and (2) a circle connecting the followingintersects: (a) V(E) R(90) I V(F) R(234); (b) V(J) R(126) I V(B) R(270);(c) V(A) R(162) I V(G) R(306); (d) V(F) R(198) I V(C) R(342); (e) V(B)R(234) I V(H) R(18); (f) V(G) R(270) I V(D) R(54); (g) V(C) R(306) IV(I) R(90); (h) V(H) R(342) I V(E) R(126), (i) V(D) R(18) I V(J) R(162);and (j) V(I) R(54) I V(A) R(198).

2. Left Iris Complex

The iris pentagon complex 150 which defines the left iris on the overlaysystem 110a is positioned on the matrix complex with V(B) of the iriscomplex 150 coinciding with intersect matrix complex V(B) R(252) Imatrix complex V(F) R(198) and iris complex V(B) R(252) coinciding withmatrix complex V(B) R(252). Features of the overlay system 110a aredefined by fragments of this pentagon complex which are the same as thefragments of the right iris complex described above.

C. Pentagon Complexes with n=5

There are five pentagon complexes 112 of size n=5 in the frontal reposeoverlay system 110a of the present invention. Three of these pentagoncomplexes 112 have Z values of 1 (FIG. 26), one has a Z value of Φ/2(FIG. 27), and one has a Z value of 2/Φ (FIG. 28). The pentagoncomplexes 112 with Z=1 are: (1) a nasal tip complex 156; (2) an internallips complex 158; and (3) an internal naris complex 160. The pentagoncomplex 112 with Z=Φ/2 is an inner nasal tip halo complex 162. Thepentagon complex 112 with Z=2/Φ is an outer nasal tip halo complex 164.These five pentagon complexes 156, 158, 160, 162, and 164 define variousfeatures of the frontal repose overlay system 110a.

1. Nasal Tip Complex

The nasal tip pentagon complex 156 (FIG. 26) is positioned on the matrixcomplex with V(A) R(180) of the complex 156 coincident with matrixcomplex V(A) R(180), nasal tip complex 156 V(C) intersected by matrixcomplex V(B) R(234), and nasal tip complex 156 V(D) intersected bymatrix complex V(E) R(126). Features of the overlay system 110a aredefined by fragments of this pentagon complex 156 comprising: (1)radials extending from each of V(F), V(G), V(I), and V(J), at 0/360° upto points where the radials respectively intersect V(F) R(216), V(B)R(252), V(E) R(108), and V(J) R(144); (2) radials connecting V(F) toV(G), V(G) to V(H), V(H) to V(I), and V(I) to V(J); (3) a radialextending from V(G) at 252° and extending to the intersect V(G) R(252) IV(A) R(180); (4) a radial extending from V(I) at 108° and extending tothe intersect V(I) R(108) I V(A) R(180); (5) a portion of a radialextending from matrix complex V(F) at 216° and extending betweenintersect matrix complex V(F) R(216) I matrix complex V(A) R(162) andintersect matrix complex V(F) R(216) I nasal tip complex V(F) R(360);and (6) a portion of a radial extending from matrix complex V(J) at 144°and extending between intersect matrix complex V(J) R(144) I matrixcomplex V(A) R(198) and intersect matrix complex V(J) R(144) I nasal tipcomplex V(J) R(360).

2. Internal Lips Complex

The internal lips pentagon complex 158 (FIG. 26) is positioned on thematrix complex with V(C) R(288) and V(D) R(72) of the internal lipscomplex 158 coincident with matrix complex V(C) R(288) and V(D) R(72)respectively. Features of the overlay system 110a are defined byfragments of this pentagon complex 158 comprising: (1) radialsconnecting V(F) to V(G), V(G) to V(C), V(C) to V(D), V(D) to V(I), andV(I) to V(J); (2) radials extending from the center point to V(B), V(E),V(F), and V(J); (3) a radial extending from V(F) at 0/360° and extendingto the intersect V(F) R(0/360) I matrix complex V(B) R(234); (4) aradial extending from V(F) at 342° and extending to the intersect V(F)R(342) I matrix complex V(B) R(234); (5) a radial extending from V(J) at0/360° and extending to the intersect V(J) R(0/360) I matrix complexV(E) R(126); and (6) a radial extending from V(J) at 18° and extendingto the intersect V(J) R(18) I matrix complex V(E). R(126). There are twoadditional fragments of this pentagon complex 158 which define featuresof the overlay system 110a. However, to describe these two additionalfeatures a nose/mouth pentagon complex 184 must first be described. Thisadditional pentagon complex 184 is of the size where n=3 and will bedescribed below. Therefore, the two additional fragments of pentagoncomplex 158 will be described concurrent with the nose/mouth pentagoncomplex 184 below.

3. Internal Naris Complex

The internal naris pentagon complex 160 (FIG. 26) is positioned on thematrix complex with V(F) R(270) of the internal naris complex 160coincident with matrix complex V(G) R(270), internal naris complex 160V(J) R(90) coincident with matrix complex V(I) R(90), and center pointof complex 160 coincident with intersect matrix complex V(B) R(234) Imatrix complex V(C) R(306). Features of the overlay system 110a aredefined by fragments of this pentagon complex 160 comprising: (1) aradial extending from V(C) at 0/360° and extending to intersect V(C)R(0/360) I V(B) R(270); (2) a radial extending from V(D) at 0/360° andextending to intersect V(D) R(0/360) I V(E) R(90); (3) a radialextending from V(F) at 234° and extending to intersect V(F) R(234) IV(J) R(126); (4) a radial extending from V(J) at 126° and extending toV(J) R(126) I V(F) R(234); (5) a radial extending from V(B) at 18° andextending to intersect point V(B) R(18) I matrix complex V(G) R(270);and (6) a radial extending from V(E) at 342° and extending to intersectpoint V(E) R(342) I matrix complex V(I) R(90). There are two additionalfragments of this pentagon complex 160 which define features of theoverlay system 110a. However, to describe these two additional fragmentsthe inner nasal tip halo pentagon complex 162 must first be described.This pentagon complex 162 is described below and the two additionalfragments of pentagon complex 160 will be described concurrently.

4. Inner Nasal Tip Halo Complex

The inner nasal tip halo pentagon complex 162 (FIG. 27) is positioned onthe matrix complex with V(G) of the inner nasal tip halo complex 162coincident with intersect matrix complex V(B) R(234) I matrix complexV(G) R(270), V(H) of the halo complex 162 coincident with intersectmatrix complex V(B) R(234) I matrix complex V(C) R(306), and V(I) of thecomplex 162 coincident with intersect matrix complex V(C) R(306) Imatrix complex V(I) R(90). Features of the overlay system 110a aredefined by fragments of this pentagon complex 162 comprising: (1)radials connecting V(Pto V(G), V(G) to V(H), V(H) to V(I), and V(I) toV(J); and (2) radials extending from V(G) and V(I) at 0/360° andextending to points where the radials respectively intersect V(F) R(90)and V(J) R(270).

Two additional fragments from the internal naris pentagon complex 160may now be defined. The additional fragments comprise: (1) a radialextending from the internal naris complex 160 V(B) at 54° and extendingto intersect internal naris complex 160 V(B) R(54) I inner nasal tiphalo complex 162 V(G) R(108); and (2) a radial extending from theinternal naris complex 160 V(E) at 306° and extending to intersectinternal naris complex 160 V(E) R(306) I inner nasal tip halo complex162 V(I) R(252).

5. Outer Nasal Tip Halo Complex

The outer nasal tip halo pentagon complex 164 (FIG. 28) is positioned onthe matrix complex with V(H) of the outer nasal tip halo complex 164coincident with intersect matrix complex V(B) R(234) I matrix complexV(C) R(306), V(G) R(54) of outer nasal tip halo complex 164 coincidentwith matrix complex V(D) R(54), and V(I) R(306) of complex 164coincident with matrix complex V(C) R(306). Features of the overlaysystem 110a are defined by fragments of this pentagon complex 164comprising radials connecting V(F) to V(G), V(G) to V(H), V(H) to V(I),and V(I) to

D. Pentagon Complexes with n=4

There are four pentagon complexes of size n=4 in the frontal reposeoverlay system 110a (see FIG. 25). All four of these pentagon complexeshave Z values of 1. The pentagon complexes are: (1) a nasal pentagoncomplex 170; (2) a chin button pentagon complex 172; (3) an eye pentagoncomplex right 174; and (4) an eye pentagon complex left 176. Fragmentsof these four pentagon complexes 170, 172, 174, and 176, define variousfeatures of the frontal repose overlay system 110a.

1. Nasal Complex

The nasal pentagon complex 170 (FIG. 25) is positioned on the matrixcomplex with V(H) of the nasal complex 170 coincident with intersectmatrix complex V(B) R(234) I matrix complex V(C) R(306), V(B) R(324) ofthe nasal complex 170 coincident with matrix complex V(C) R(324), andV(E) R(216) of the nasal complex 170 coincident with matrix complex V(F)R(216). Various features of the overlay system 110a are defined byfragments of this pentagon complex 170 comprising: (1) a radialextending from V(F) at 180° and extending to intersect V(F) R(180) IV(G) R(252); (2) a radial extending from V(J) at 180° and extending tointersect V(J) R(180) I V(I) R(108); (3) a radial connecting V(F) toV(G); (4) a radial connecting V(J) to V(I); (5) a portion of radial V(B)R(234) which connects the radials described in (1) and (3) above; (6) aportion of radial V(E) R(126) which connects the radials described in(2) and (4) above; (7) a radial extending from V(G) at 18° and extendingto intersect V(G) R(18) I V(B) R(36); and (8) a radial extending fromV(I) at 342° and extending to intersect V(I) R(342) I V(E) R(324).

In additions two fragments of the nasal pentagon complex 170 can bedescribed if the inner nasal tip halo pentagon complex 162 is alsoconsidered. These additional fragments comprise: (9) a radial extendingfrom nasal complex 170 V(G) at 180° and extending to intersect nasalcomplex 170 V(G) R(180) I inner nasal tip halo complex 162 V(G) R(108);and (10) a radial extending from nasal complex 170 V(I) at 180° andextending to intersect nasal complex 170 V(I) R(252).

2. Chin Button Complex

The chin button pentagon complex 172 (FIG. 25) is positioned on thematrix complex with V(H) of the chin button complex 172 coincident withmatrix complex V(H), V(B) of the chin complex 172 coincident withintersect matrix complex V(B) R(216) I matrix complex V(C) R(270), andV(E) of the complex 172 coincident with matrix complex V(D) R(90) I V(E)R(144). Features of the frontal repose overlay system 110a are definedby fragments of this pentagon complex 172 comprising: (1) radialsconnecting V(E) to V(J), V(J) to V(A), V(A) to V(F), and V(F) to V(B);(2) a radial extending from V(B) at 198° and extending to intersect V(B)R(198) I V(G) R(270); (3) a radial extending from V(E) at 162° andextending to intersect V(E) R(162) I V(I) R(90); (4) a radial extendingfrom V(B) at 54° and extending to intersect V(B) R(54) I V(C) R(36); (5)a radial extending from V(E) at 306° and extending to intersect V(E)R(306) I V(D) R(324); (6) a portion of radial V(G) R(270) which connectsthe radials described in (2) and (3) above; (7) a portion of radial V(C)R(324) which connects a center point of complex 172 and the radialdescribed in (6) above; and (8) a portion of radial V(D) R(36) whichconnects the center point of the complex 172 to the radial described in(6) above.

3. Right Eve Complex

The right eye pentagon complex 174 (FIG. 25) is positioned on the matrixcomplex with V(E) R(108) of the eye complex 174 coincident with V(E)R(108) of the matrix complex and intersect V(I) R(54) I V(J) (162) ofthe eye pentagon complex 174 coincident with intersect matrix complexV(I) R(54) I matrix complex V(J) R(162). Various features of the frontalrepose overlay system 110a are defined by fragments of this pentagoncomplex 174 comprising: (1) radials connecting V(G) to V(C), V(C) toV(H), V(H) to V(D), V(D) to V(I), and V(I) to V(E); (2) a radialconnecting V(B) to V(J); (3) a radial extending from V(G) at 252° andextending to intersect V(G) R(252) I V(H) R(0/360); (4) a radialextending from V(I) at 108° and extending to intersect V(I) R(108) IV(H) R(0/360); (5) a radial extending from V(G) at 270° and extending tointersect V(G) R(270) I V(E) R(162); (6) a radial extending from V(G) at306° and extending to intersect V(G) R(306) I V(F) R(216); (7) a radialextending from V(G) at 324° and extending to intersect V(G) R(324) IV(B) R(270); (8) a radial extending from V(G) at 342° and extending tointersect V(G) R(342) I V(B) R(216); (9) a line connecting intersectV(G) R(342) I V(B) R(216) and intersect V(G) R(324) I V(B) R(270); (10)a radial extending from V(E) at 90° and extending to intersect V(E)R(90) I V(G) R(324); (11) a portion of a radial extending from V(E) at162° and extending between intersect V(E) R(162) I V(G) R(270) and V(E)R(162) I V(I) R(36); (12) a portion of a radial extending from V(I) at36° and extending between intersect V(I) R(36) I V(E) R(162) andintersect V(I) R(36) I V(E) R(108); (13) a line connecting intersectV(I) R(36) I V(E) R(108) and intersect V(A) R(198) I V(B) R(270); (14) aportion of a radial extending from V(E) at 144° and extending betweenintersect V(E) R(144) I V(F) R(234) and V(E) R(144) I V(G) R(270); (15)a portion of a radial extending from V(I) at 54° and extending betweenintersect V(I) R(54) I V(A) R(198) and intersect V(I) R(54) I V(E)R(144); (16) a radial extending from V(G) at 234° and extending tointersect V(G) R(234) I V(C) R(270); (17) a portion of a radialextending from V(C) at 270° and extending between intersect V(C) R(270)I V(G) R(234) and intersect V(C) R(270) I V(I) R(126); (18) a radialextending from V(I) at 126° and extending to intersect V(I) R(126) IV(C) R(270); and (19) a radial extending from V(B) at 36° and extendingto intersect V(B) R(36) I matrix complex V(J) R(144).

4. Left Eve Complex

The left eye pentagon complex 176 (FIG. 25) is positioned on the matrixcomplex with V(B) R(252) of the eye complex 176 coincident with V(B)R(252) of the matrix complex and intersect V(F) R(198) I V(G) R(306) ofthe eye complex 176 coincident with intersect V(F) R(198) I V(G) R(306)of the matrix complex. Various features of the frontal repose overlaysystem 110a are defined by fragments of this pentagon complex 176comprising: (1) radials connecting V(B) to V(G), V(G) to V(C), V(C) toV(H), V(H) to V(D), and V(C) to V(I); (2) a radial connecting V(E) toV(F); (3) a radial extending from V(G) at 252° and extending tointersect V(G) R(252) I V(H) R(0/360); (4) a radial extending from V(I)at 108° and extending to intersect V(I) R(108) I V(H) R(0/360); (5) aradial extending from V(I) at 270° and extending to intersect V(I)R(270) I V(B) R(198); (6) a radial extending from V(I) at 54° andextending to intersect V(I) R(54) I V(J) R(144); (7) a radial extendingfrom V(I) at 36° and extending to intersect V(I) R(36) I V(B) R(270);(8) a radial extending from V(I) at 18° and extending to intersect V(I)R(18) I V(E) R(144); (9) a line connecting intersect V(I) R(18) I V(E)R(144) and intersect V(I) R(36) I V(B) R(270); (10) a radial extendingfrom V(B) at 270° and extending to intersect V(B) R(270) I V(I) R(36);(11) a portion of a radial extending from V(B) at 198° and extending.between intersect V(B) R(198) I V(I) R(90) and V(B) R(198) I V(G)R(324); (12) a portion of a radial extending from V(G) at 324° andextending between intersect V(G) R(324) I V(B) R(198) and intersect V(G)R(324) I V(B) R(252); (13) a line connecting intersect V(G) R(324) IV(B) R(252) and intersect V(A) R(162) I V(B) R(270); (14) a portion of aradial extending from V(B) at 216° and extending between intersect V(B)R(216) I V(I) R(270) and intersect V(B) R(216) I V(J) R(126); (15) aportion of a radial extending from V(G) at 306° and extending betweenintersect V(G) R(306) I V(A) R(162) and intersect V(G) R(306) I V(B)R(216); (16) a radial extending from V(G) at 234° and extending tointersect V(G) R(234) I V(C) R(270); (17) a portion of a radialextending from V(C) at 270° and extending between intersect V(C) R(270)I V(G) R(234) and intersect V(C) R(270) I V(I) R(126); (18) a radialextending from V(I) at 126° and extending to intersect V(I) R(126) IV(C) R(270); and (19) a radial extending from V(E) at 324° and extendingto intersect V(E) R(324) I matrix complex V(F) R(216).

E. Pentagon Complexes with n=3

A majority of the pentagon complexes 112 making up the frontal reposeoverlay system 110a of the present invention are of the size n=3. Thereare sixteen pentagon complexes of this size. Fourteen of these complexeshave Z values of 1 (See FIG. 22). These pentagon complexes are: (1) anose/mouth pentagon complex 184; (2) a mouth/chin pentagon complex 186;(3) a chin inferior border pentagon complex 188; (4) a chin pentagoncomplex 190; (5) a right side chin pentagon complex 192; (6) a left sidechin pentagon complex 194; (7) a right eye/cheek pentagon complex 196;(8) a left eye/cheek pentagon complex 198; (9) a right eye brow pentagoncomplex 200; (10) a left eye brow pentagon complex 202; (11) a rightcheek pentagon complex 204; (12) a left cheek pentagon complex 206; (13)a right side nose/mouth pentagon complex 208; and (14) a left sidenose/mouth pentagon complex 210. The remaining two complexes have Zvalues of (Φ)^(1/3) (See FIG. 23). These pentagon complexes are: (15) aright eye brow/cheek pentagon complex 212; and (16) a left eyebrow/cheek pentagon complex 214. Fragments of these sixteen pentagoncomplexes 184, 186, 188, 190, 192, 194, 196, 198, 200, 202, 204, 206,208, 210, 212, and 214, define various features of the overlay system110a of the present invention.

1. Nose/Mouth Complex

The nose/mouth pentagon complex 184 (FIG. 22) is positioned on thematrix complex with V(I) and V(F) of the nose/mouth complex 184coincident to matrix complex radial V(B) R(234), V(G) and V(J) of thenose/mouth complex 184 coincident to matrix complex radial V(E) R(126),V(G) coincident with intersect matrix complex V(C) R(306) I matrixcomplex V(D) R(72), and V(I) coincident with intersect matrix complexV(C) R(288) I matrix complex V(D) R(54). Various features of the overlaysystem 110a are defined by fragments of this pentagon complex 184comprising: (1) a radial extending between V(G) and V(F); (2) a radialextending between V(I) and V(J); (3) a radial extending from V(G) at324° and extending to intersect V(G) R(324) I V(F) R(216); (4) a radialextending from V(I) at 36° and extending to intersect V(I) R(36) I V(J)R(144); (5) a radial extending from V(G) at 72° and extending tointersect V(G) R(72) I V(B) R(126); (6) a radial extending from V(I) at288° and extending to intersect V(I) R(288) I V(E) R(234); (7) a radialextending from V(G) at 126° and extending to intersect V(G) R(126) IV(H) R(72); (8) a radial extending from V(I) at 234° and extending tointersect V(I) R(234) I V(H) R(288); (9) a radial extending from V(G) at234° and extending to intersect V(G) R(234) I V(C) R(306); (10) a radialextending from V(I) at 126° and extending to intersect V(I) R(126) IV(D) R(54); (11) a line connecting intersect V(G) R(234) I V(C) R(306)and intersect V(I) R(126) I V(D) R(54); (12) a portion of a radialextending from V(I) at 108° and extending between intersect V(I) R(108)I V(G) R(180) and intersect V(I) R(108) I matrix complex V(H) R(18);(13) a portion of a radial extending from V(G) at 252° and extendingbetween intersect V(G) R(252) I V(I) R(180) and V(G) R(252) I matrixcomplex V(H) R(342); (14) a radial extending from V(G) at 270° andextending to intersect V(G) R(270) I V(H) R(36); (15) a radial extendingn from V(I) at 90° and extending to intersect V(I) R(90) I V(H) R(324);(16) a portion of a radial extending from matrix complex V(G) at 270°and extending between intersect matrix complex V(G) R(270) I matrixcomplex V(F) R(216) and intersect matrix complex V(G) R(270) Inose/mouth complex V(I) R(324); and (17) a portion of a radial extendingfrom matrix complex V(G) at 270° and extending between intersect matrixcomplex V(G). R(270) I matrix complex V(F) R(198) and intersect matrixcomplex V(G) R(270) I nose/mouth complex V(G) R(36).

Taking into consideration the internal lips pentagon complex 158(described above), four additional fragments may be described asfollows: (18) a radial extending from V(G) of the nose/mouth pentagoncomplex 184 at 288° and extending to V(F) of the internal lips pentagoncomplex 158; (19) a radial extending from V(I) of the nose/mouthpentagon complex 184 at 72° and extending to V(J) of the internal lipspentagon complex 158; (20) a radial extending from V(B) of the internallips pentagon complex 158 at 108° and extending to intersect internallips pentagon complex 158 V(B) R(108) I nose/mouth pentagon complex 184V(G) R(270); and (21) a radial extending from V(E) at 252° and extendingto intersect internal lips pentagon complex 158 V(E) R(252) I nose/mouthpentagon complex 184 V(I) R(90).

2. Mouth/Chin Complex

The mouth/chin pentagon complex 186 (FIG. 22) is positioned on thematrix complex with V(A) of the mouth/chin complex 186 coincident withintersect matrix complex V(B) R(234) I matrix complex V(E) R(126), V(C)of the mouth/chin complex 186 coincident with intersect matrix complexV(B) R(216) I V(C) R(270), and V(D) of the mouth/chin complex coincidentwith intersect matrix complex V(E) R(144)I V(D) R(90). Features of theoverlay system 110a are defined by fragments of this pentagon complex186 comprising: (1) a radial connecting V(G) and V(I); (2) a radialextending from V(G) at 36° and extending to intersect V(G) R(36) I V(B)R(126); and (3) a radial extending from V(I) at 324° and extending tointersect V(I) R(324) I V(E) R(234).

3. Chin Inferior Border Complex

The chin inferior border pentagon complex 188 (FIG. 22) is positioned onthe matrix pentagon complex with V(A) R(180) of the border complex 188coincident with V(A) R(180) of the matrix complex and chin complex 188intersect V(F) R(234) I chin complex 188 V(J) R(126) coincident withintersect matrix complex V(C) R(288) I matrix complex V(D) R(72).Features of the overlay system 110a are defined by fragments of thispentagon complex 198 comprising: (1) a radial connecting V(C) and V(D);(2) a radial extending from V(C) at 54° and extending to intersect V(C)R(54) I V(B) R(126); (3) a radial extending from V(D) at 306° andextending to intersect V(D) R(306) I V(E) R(234); (4) a radial extendingfrom V(C) at 18° and extending to the intersect V(C) R(18) I V(G)R(252); and (5) a radial extending from V(D) at 342° and extending tothe intersect V(D) R(342) I V(I) R(108).

4. Chin Complex

The chin pentagon complex 190 (FIG. 22) is positioned on the matrixcomplex with V(B) of the chin complex 190 coincident with intersectmatrix complex V(B) R(216) I matrix complex V(C) R(288), V(E) of thechin complex 190 coincident with intersect matrix complex V(D) R(72) Imatrix complex V(E) R(144), and V(A) R(180) of the chin complexcoincident with V(A) R(180) of the matrix complex. Features of thefrontal repose overlay system 110a are defined by fragments of thispentagon complex 190 comprising: (1) a radial connecting V(B) and V(E);(2) a radial extending from V(B) at 144° and extending to intersect V(B)R(144) I V(G) R(18); (3) a radial extending from V(E) at 216° andextending to intersect V(E) R(216) I V(I) R(342); (4) a radial extendingfrom V(G) at 18° and extending to intersect V(G) R(18) I V(B) R(144);(5) a radial extending from V(I) at 342° and extending to intersect V(I)R(342) I V(E) R(216); (6) a portion of a radial extending from V(A) at144° and extending between intersect V(A) R(144) I V(B) R(270) andintersect V(A) R(144) I V(B) R(198); (7) a portion of a radial extendingfrom V(A) at 216° and extending between intersect V(A) R(216) I V(E)R(90) and intersect V(A) R(216) I V(E) R(162); (8) a portion of a radialextending from V(B) at 198° and extending between intersect V(B) R(198)I V(G) R(324) and intersect V(B) R(198) I V(G) R(234); (9) a portion ofa radial extending from V(E) at 162° and extending between intersectV(E) R(162) I V(I) R(36) and intersect V(E) R(162) I V(I) R(126); (10) aportion of a radial-extending from V(G) at 234° and extending betweenintersect V(G) R(234) I V(C) R(18) and intersect V(G) R(234) I V(C)R(342); (11) a portion of a radial extending from V(I) at 126° andextending between intersect V(I) R(126) I V(D) R(342) and intersect V(I)R(126) I V(D) R(18); (12) a portion of a radial extending from V(C) at306° and extending between intersect V(C) R(306) I V(D) R(54) andintersect V(C) R(306) I V(D) R(72); and (13) a portion of a radialextending from V(D) at 54° and extending between intersect V(D) R(54) IV(C) R(306) and intersect V(D) R(54) I V(C) R(288).

5. Right Side Chin Complex

The right side chin pentagon complex 192 (FIG. 22) is positioned on thematrix pentagon complex with V(A) of the chin complex 192 coincidentwith matrix complex intersect matrix complex V(B) R(234) I matrixcomplex V(C). R(288) and V(B) R(180) of the chin complex 192 coincidentwith V(A) R(180) of the matrix complex. Features of the overlay system110a are defined by fragments of this pentagon complex 192 including aportion of a radial extending from V(I) of the matrix pentagon complexat 324° and extending between intersect matrix complex V(I) R(324) Ichin complex 192 V(B) R(198) and intersect matrix complex V(I) R(324) Ichin complex 192 V(B) R(234).

6. Left Side Chin Complex

The left side chin pentagon complex 194 (FIG. 22) is positioned on thematrix pentagon complex with V(A) of the chin complex 194 coincidentwith intersect matrix complex V(C) R(306) I matrix complex V(D) R(72)and V(E) R(180) of the chin complex 194 coincident with V(A) R(180) ofthe matrix complex. Features of the overlay system 110a are defined byfragments of this pentagon complex 194 including a portion of a radialextending from V(I) of the matrix pentagon complex at 36° and extendingbetween intersect matrix complex V(I) R(36) I chin complex 194 V(E)R(126) and intersect matrix complex V(I) R(36) I chin complex 194 V(E)R(162).

7. Right Eye/Cheek Complex

The right eye/cheek pentagon complex 196 (FIG. 22) is positioned on thematrix pentagon complex with V(B) R(108) of the eye/cheek complex 196coincident to V(E) R(108) of the matrix complex, V(G) R(216) of theeye/cheek complex 196 coincident to V(F) R(216) of the matrix complex,and V(J) R(288) of the eye/cheek complex 196 coincident to V(G) R(288)of the matrix complex. Features of the overlay system 110a are definedby fragments of this pentagon complex 196 comprising: (1) a radialconnecting V(B) and V(C); (2) a radial extending from V(B) at 216° andextending to intersect V(B) R(216) I V(C) R(342); (3) a radial extendingfrom V(B) at 234° and extending to intersect eye/cheek complex V(B)R(234) I matrix complex V(I) R(342); (4) a portion of a radial extendingfrom V(B) at 252° and extending between intersect V(B) R(252) I V(I)R(18) and intersect V(B) R(252) I V(I) R(0/360); (5) a portion of aradial extending from V(E) at 108° and extending between intersect V(E)R(108) I V(I) R(0/360) and intersect V(E) R(108) I V(I) R(342); (6) aradial extending from V(E) at 54° and extending to intersect V(E) R(54)I V(J) R(198); (7) a radial connecting V(E) and V(I); (8) a radialconnecting V(I) and V(D); (9) a radial connecting V(J) and V(A); (10) aradial extending from V(A) at 126° and extending to intersect V(A)R(126) I V(F) R(162); (11) a portion of a radial extending from V(F) at162° and extending between intersect V(F) R(162) I V(B) R(306) andintersect V(F) R(162) I V(B) R(270); (12) a portion of a radialextending from V(B) at 270° and extending between intersect V(B) R(270)I V(F) R(162) and intersect V(B) R(270) I V(F) R(180); (13) a portion ofa radial extending from V(J) at 144° and extending between intersectV(J) R(144) I V(I) R(36) and intersect eye/cheek complex 196 V(J) R(144)I matrix complex V(I) R(54); and (14) a portion of a radial extendingfrom matrix complex V(I) at 54° and extending between intersect matrixcomplex V(I) R(54) I eye/cheek complex 196 V(J) R(144) and intersectmatrix complex V(I) R(54) I eye/cheek complex 196 V(J) R(126). Anadditional feature may be described by taking into account the chinbutton complex 172 described above, this additional feature comprises:(15) a radial extending from eye/cheek complex 196 V(E) at 162° andextending to intersect eye/cheek complex 196 V(E) R(162) I chin buttoncomplex 172 V(E) R(306). When considering this complex in conjunctionwith the nose/mouth pentagon complex described above, additionalfragments may be described comprising: (16) a radial extending fromnose/mouth complex V(I) at 324° and extending to intersect nose/mouthcomplex V(I) R(324) I eye/cheek complex V(D).

8. Left Eve/Cheek Complex

The left eye/cheek pentagon complex 198 (FIG. 22) is positioned on thematrix pentagon complex with V(E) R(252) of the eye/cheek complex 198coincident to V(B) R(252) of the matrix complex, V(I) R(144) of theeye/cheek complex 198 coincident to V(J) R(144) of the matrix complex,and V(F) R(72) of the eye/cheek complex 198 coincident to V(I) R(72) ofthe matrix complex. Features of the overlay system 110a are defined byfragments of this pentagon complex 198 comprising: (1) a radialconnecting V(E) and V(D); (2) a radial extending from V(E) at 144° andextending to intersect V(E) R(144) I V(D) R(18); (3) a radial extendingfrom V(E) at 126° and extending to intersect eye/cheek complex 198 V(E)R(126) I matrix complex V(G) R(18); (4) a portion of a radial extendingfrom V(E) at 108° and extending between intersect V(E) R(108) I V(G)R(342) and intersect V(E) R(108) I V(G) R(0/360); (5) a portion of aradial extending from V(B) at 252° and extending between intersect V(B)R(252) I V(G) R(0/360) and intersect V(B) R(252) I V(G) R(18); (6) aradial extending from V(B) at 306° and extending to intersect V(B)R(306) I V(F) R(162); (7) a radial connecting V(B) and V(G); (8) aradial connecting V(G) and V(C); (9) a radial connecting V(F) and V(A);(10) a radial extending from V(A) at 234° and extending to intersectV(A) R(234) I V(J) R(198); (11) a portion of a radial extending fromV(J) at 198° and extending between intersect V(J) R(198) I V(E) R(54)and intersect V(J) R(198) I V(E) R(90); (12) a portion of a radialextending from V(E) at 90° and extending between intersect V(E) R(90) IV(J) R(198) and intersect V(E) R(90) I V(J) R(180); (13) a portion of aradial extending from V(F) at 216° and extending between intersect V(F)R(216) I V(G) R(324) and intersect eye/cheek 1% complex 198 V(F) R(216)I matrix complex V(G) R(306); and (14) portion of a radial extendingfrom matrix complex V(G) at 306° and extending between intersect matrixcomplex V(G) R(306) I eye/cheek complex 198 V(F) R(216) and intersectmatrix complex V(G) R(306) I eye/cheek complex 198 V(F) R(234). Anadditional feature may be described by taking into account the chinbutton complex 172 described above, this additional feature comprises:(15) a radial extending from eye/cheek complex 198 V(B) at 198° andextending to intersect eye/cheek complex 198 V(B) R(198) I chin buttoncomplex 172 V(B) R(54). When considering this complex in conjunctionwith the nose/mouth pentagon complex described above, additionalfragments may be described comprising: (16) a radial extending fromnose/mouth complex V(G) at 36° and extending to intersect nose/mouthcomplex V(G) R(36) I eye/cheek complex V(C);

9. Right Eye Brow Complex

The right eye brow pentagon complex 200 (FIG. 22) is positioned on thematrix pentagon complex with radial V(E) R(108) coincident with matrixcomplex V(E) R(108) and matrix complex V(F) R(234) intersecting eye browcomplex 200 V(A) R(144) I eye brow complex 200 V(B) R(288). Variousfeatures of the overlay system 110a are defined by fragments of thispentagon complex 200 comprising: (1) a radial extending from V(G) at324° and extending to intersect V(G) R(324) I V(B) R(252); (2) a radialextending from V(J) at 234° and extending to intersect V(J) R(234) Imatrix complex V(J) R(180); (3) a radial extending from V(J) at 108° andextending to intersect V(J) R(108) I V(F) R(162); (4) a radialconnecting intersect matrix complex V(A) R(216) I matrix complex V(J)R(162) and intersect matrix complex V(A) R(198) I matrix complex V(B)R(270); (5) a point at intersect V(E) R(216) I V(I) R(270); and (6) apoint at intersect V(I) R(198) I V(D) R(252).

10. Left Eve Brow Complex

The left eye brow pentagon complex 202 (FIG. 22) is positioned on thematrix pentagon complex with radial V(B) R(252) coincident with matrixcomplex V(B) R(252) and matrix complex V(J) R(126) intersecting eye browcomplex 202 V(A) R(216) I eye brow complex 202 V(E) R(72). Variousfeatures of the overlay system 110a are defined by fragments of thispentagon complex 202 comprising: (1) a radial extending from V(I) at 36°and extending to intersect V(I) R(36) I V(E) R(108); (2) a radialextending from V(F) at 126° and extending to intersect V(F) R(126) Imatrix complex V(F) R(180); (3) a radial extending from V(F) at 252° andextending to intersect V(F) R(252) I V(J) R(198); (4) a radialconnecting intersect matrix complex V(A) R(144) I matrix complex V(F)R(198) and intersect matrix complex V(A) R(162) I matrix complex V(E)R(90); (5) a point at intersect V(B) R(144) I V(G) R(90); and (6) apoint at intersect V(G) R(162) I V(C) R(108).

11. Right Cheek Complex

The right cheek pentagon complex 204 (FIG. 22) is positioned on thematrix pentagon complex with V(A) of the cheek complex 204 coincidentwith intersect matrix complex V(A) R(198) I matrix complex V(B) R(252)and V(I) of the cheek complex 204 coincident with intersect matrixcomplex V(C) R(288) I matrix complex V(D) R(0/360). Features of theoverlay system 110a are defined by fragments of this pentagon complex204 comprising: (1) a radial connecting V(B) and V(C) of the cheekcomplex 204; (2) a radial connecting V(E) and V(D) of the cheek complex204; and (3) a radial extending from V(D) at 144° and extending tointersect cheek complex 204 V(D) R(144) I matrix complex V(H) R(306).

12. Left Cheek Complex

The left cheek pentagon complex 206 (FIG. 22) is positioned on thematrix complex with V(A) of the cheek complex 206 coincident withintersect matrix complex V(A) R(162) I matrix complex V(E) R(108) andV(G) of the cheek complex 206 coincident with intersect matrix complexV(D) R(72) I matrix complex V(C) R(0/360). Features of the overlaysystem 110a are defined by fragments of this pentagon complex 206comprising: (1) a radial connecting V(D) and V(E) of the cheek complex206; (2) a radial connecting V(B) and V(C) of the cheek complex 206; and(3) a radial extending from V(C) at 216° and extending to intersectcheek complex 206 V(C) R(216) I matrix complex V(H) R(180).

13. Right Side Nose/Mouth Complex

The right side nose/mouth pentagon complex 208 (FIG. 22) is positionedon the matrix pentagon complex with V(F) of the nose/mouth complex 208coincident with V(C) R(306) of the matrix complex and nose/mouth complex208 V(I) R(198) coincident with matrix complex V(A) R(198). Features ofthe overlay system 110a are defined by fragments of this pentagoncomplex 208 comprising: (1) a radial extending from V(J) at 144° andextending to intersect V(J) R(144) I V(F) R(198); and (2) a portion of aradial extending from V(C) at 306° and extending between intersect V(C)R(306) I V(A) R(198) and intersect V(C) R(306) I V(I) R(342).

14. Left Side Nose/Mouth Complex

The left side nose/mouth pentagon complex 210 (FIG. 22) is positioned onthe matrix pentagon complex with V(J) of the nose/mouth complex 210coincident with V(D) R(54) of the matrix complex and nose/mouth complex210 V(G) R(162) coincident with matrix complex V(A) R(162). Features ofthe overlay system 110a are defined by fragments of this pentagoncomplex 210 comprising: (1) a radial extending from V(F) at 216° andextending to intersect V(F) R(216) I V(J) R(162); and (2) a portion of aradial extending from V(D) at 54° and extending between intersect V(D)R(54) I V(A) R(162) and intersect V(D) R(54) I V(G) R(18).

15. Right Eye Brow/Cheek Complex

The right eye brow/cheek pentagon complex 212 (FIG. 23) is positioned onthe matrix pentagon complex with an 180 clockwise rotation with V(E) ofthe brow/cheek complex coincident to intersect matrix complex V(E)R(108) I matrix complex V(I) R(36) and brow/cheek complex 212.V(E)R(162) coincident to matrix complex V(D) R(0/360). Features of theoverlay system 110a are defined by fragments of this pentagon complex212 comprising: (1) radials connecting V(H) to V(I), V(I) to V(E), V(E)to V(J), and V(J) to V(F); (2) a radial connecting V(H) to V(D); (3) aradial extending from V(I) at 144° and extending to intersect V(I)R(144) I V(H) R(252); (4) a radial extending from V(E) at 126° andextending to intersect V(E) R(126) I V(A) R(198); (5) a radial extendingfrom V(E) at 108° and extending to intersect V(E) R(108) I V(A) R(198);(6) a radial extending from V(E) at 72° and extending to intersect V(E)R(72) I V(A) R(180); (7) a portion of a radial extending from V(J) at108° and extending between intersect V(J) R(108) I V(A) R(180) andintersect V(J) R(108) I V(F) R(162); (8) a radial extending from V(E) at54° and extending to intersect V(E) R(54) I V(J) R(144); and (9) aportion of a radial extending from matrix complex V(E) at 144° andextending between intersect matrix complex V(E) R(144) I matrix complexV(I) R(108) and intersect matrix complex V(E) R(144) I eye brow/cheekcomplex V(D) R(144). Additional fragments of the complex 212 may bedescribed by taking into account the right eye complex 174 describedabove. The additional fragments comprise: (10) a portion of a radialextending. from V(I) of the brow/cheek complex 212 at 72° and extendingbetween intersect brow/cheek complex 212 V(I) R(72) I brow/cheek complex212 V(A) R(198) and intersect brow/cheek complex 212 V(I) R(72) I eyecomplex 174 V(D) R(342); and (11) a radial extending from eye complexV(B) at 288° and extending to intersect eye complex V(B) R(288) I eyebrow/cheek complex V(E) R(54).

Furthermore, when considering this complex in conjunction with the rightcheek pentagon complex described above, additional fragments may bedescribed comprising: (12) a portion of a radial extending from matrixcomplex V(J) at 180° and extending between intersect matrix complex V(J)R(180) I eye brow/cheek complex V(D) and intersect matrix complex V(J)R(180) I cheek complex V(E).

16. Left Eye Brow/Cheek Complex

The left eye brow/cheek pentagon complex 214 (FIG. 23) is positioned onthe matrix pentagon complex with an 18° clockwise rotation with V(F) ofthe brow/cheek complex coincident to intersect matrix complex V(A)R(144) I matrix complex V(F) R(180) and brow/cheek complex (F) R(126)coincident to matrix complex V(A) R(144). Features of the overlay system110a are defined by fragments of this pentagon complex comprising: (1)radials connecting V(C) to V(B), V(B) to V(F), V(F) to V(A), and V(A) toV(E); (2) a radial connecting V(C) to V(G); (3) a radial extending fromV(B) at 180° and extending to intersect V(B) R(180) I V(C) R(72); (4) aradial extending from V(C) at 0/360° and extending to intersect V(C)R(0/360) I V(B) R(252); (5) a radial extending from V(F) at 270° andextending to intersect V(F) R(270) I V(A) R(180); (6) a radial extendingfrom V(F) at 252° and extending to intersect V(F) R(252) I V(A) R(216);(7) a radial extending from V(F) at 216° and extending to intersect V(F)R(216) I V(J) R(126); (8) a radial extending from V(F) at 198° andextending to intersect V(F) R(198) I V(J) R(126); (9) a radial extendingfrom V(I) at 144° and extending to intersect V(I) R(144) I V(H) R(252);(10) a radial extending from V(I) at 126° and extending to intersectV(I) R(126) I V(D) R(54); (11) a radial extending from V(I) at 108° andextending to intersect V(I) R(108) I V(D) R(54); (12) a portion of aradial extending from V(I) at 36° and extending between intersect V(I)R(36) I V(E) R(162) and intersect V(I) R(36) I V(E) R(72); (13) aportion of a radial extending from matrix complex V(B) at 216° andextending between intersect matrix complex V(B) R(216) I matrix complexV(G) R(252) and intersect matrix complex V(B) R(216) I eye brow/cheekcomplex V(B) R(180). Additional features may be described by taking intoaccount the left eye complex described above. The additional featurescomprise: (14) a portion of a radial extending from V(I) of thebrow/cheek complex at 90° and extending between intersect brow/cheekcomplex V(I) R(90) I brow/cheek complex V(E) R(126) and intersectbrow/cheek complex V(I) R(90) I eye complex V(D) R(54); and (15) aradial extending from eye complex V(E) at 72° and extending to intersecteye complex V(E) R(72) I eye brow/cheek complex V(F) R(270).

Furthermore, when considering this complex in conjunction with the leftcheek pentagon complex described above, additional fragments may bedescribed comprising: (16) a portion of a radial extending from matrixcomplex V(F) at 180° and extending between intersect matrix complex V(F)R(180) I eye brow/cheek complex V(G) and intersect matrix complex V(F)R(180) I cheek complex V(B).

F. Pentagon Complexes with n=2

Only one of the pentagon complexes 112 making up the frontal reposeoverlay system 110a is of the size n=2 and Z=1 (See FIG. 18). Thispentagon complex is a frontal repose smile pentagon complex 220.

1. Repose Smile Complex

The repose smile pentagon complex 220 (FIG. 18) is positioned on thematrix pentagon complex with V(B) of the repose smile complex 220coincident with matrix pentagon complex intersect matrix complex V(B)R(216) I matrix complex V(C) R(324) and V(E) of the repose smile complex220 coincident with matrix pentagon complex intersect matrix complexV(E) R(144) I matrix complex V(D) R(36). Fragments of this pentagoncomplex 220 define features of the frontal repose overlay system 110acomprising: (1) a radial connecting V(J) and V(E); and (2) a radialconnecting V(F) and V(B). Additional fragments of this pentagon complex220 may be used to define a more prominent lower jaw for use in anoverlay system 110 to be used for analyzing male facial configurationsas mentioned above.

G. Pentagon Complexes with n=1

Only one of the pentagon complexes 112 making up the frontal reposeoverlay system 110a is of the size n=1 and Z=1 (See FIG. 17). Thispentagon complex is an internal facial pentagon complex 222.

1. Internal Facial Complex

The internal facial pentagon complex 222 (FIG. 17) is positioned on thematrix pentagon complex with V(A) R(180) of the internal facial complex222 coincident with V(A) R(180) of the matrix complex and V(H) of theinternal facial complex is coincident with V(H) of the matrix complex.Fragments of this I pentagon complex 222 define features of the frontalrepose overlay system 110a. All of the features defined by the complex222 have been defined above in various smaller pentagon complexes.

II. Lateral Repose Overlay System

A overlay system 110b comprising a lateral view of a face in repose(FIG. 9B) will be described by following the same procedure as usedabove to describe the frontal repose overlay system 110a: describing thematrix complex 134 first, then the smallest pentagon complexes, andwithin a particular size of complex describing its position with respectto a matrix pentagon complex 134 and then describing points, lines,and/or areas defined by the complex being described which outline theoverlay system 110b. As in the overlay system 110a, there are sevendifferent sizes of pentagon complexes involved in the overlay system110b comprising a lateral view of a face in repose.

A. Pentagon Complexes with n=0

There is only one pentagon complex of the size n=0 included in thelateral repose overlay system 110b (See FIG. 16). This pentagon complexhas a Z value of 1 and is called a lateral matrix pentagon complex 134b(See FIG. 16).

1. Lateral Matrix Complex

Features of the lateral repose overlay system 110b are defined byfragments of the lateral matrix pentagon complex 134b comprising: (1) aportion of a radial extending from V(A) at 234° and extending betweenintersect V(A) R(234) I V(J) R(90) and intersect V(A) R(234) I V(J)R(126); (2) a line connecting intersect V(A) R(234) I V(J) R(126) andintersect V(J) R(144) I V(A) R(216); (3) a portion of a radial extendingfrom V(A) at 216° and extending between intersect V(A) R(216) I V(J)R(144) and intersect V(A) R(216) I V(J) R(162); (4) a line connectingintersect V(J) R(162) I V(A) R(216) and intersect V(E) R(90) I V(J)R(180); (5) a line connecting intersect V(E) R(90) I V(J) R(180) andintersect V(A) R(216) I V(E) R(126); (6) a line extending from intersectV(E) R(126) I V(A) R(216) at 270° and extending to the intersect of thatline and V(E) R(162); (7) a radial extending from V(E) at 162° andextending to intersect V(E) R(162) I V(I) R(108); (8) a portion of aradial extending from V(E) at 144° and extending between intersect V(E)R(144) I V(J) R(198) and intersect V(E) R(144) I V(I) R(108); (9) aportion of a radial extending from V(E) at 126° and extending betweenintersect V(E) R(126) I V(I) R(36) and intersect V(E) R(126) I V(H)R(342); (10) a portion of a radial extending from V(E) at 108° andextending between intersect V(E) R(108) and the line described in (4)above and intersect V(E) R(108) I V(H) R(342); (11) a portion of aradial extending from V(J) at 162° and extending between intersect V(J)R(162) I V(I) R(54) and intersect V(J) R(162) I V(I) R(90); (12) aportion of a radial extending from V(D) at 36° and extending betweenintersect V(D) R(36) I V(H) R(324) and intersect V(D) R(36) I V(H)R(342); (13) a portion of a radial extending from V(I) at 126° andextending between intersect V(I) R(126) I V(D) R(342) and intersect V(I)R(126) I V(D) R(54); (14) a radial extending from V(I) at 180° andextending to intersect V(I) R(180) I V(D) R(306); (15) a radialextending from V(D) at 306° and extending to intersect V(D) R(306) IV(I) R(180); (16) a portion of a radial extending from V(F) at 351° andextending between intersect V(F) R(351) I V(B) R(270) and intersect V(F)R(351) I V(F) R(270); (17) a radial extending from V(D) at 288° andextending to intersect V(D) R(288) I V(E) R(166.5); (18) a lineextending between intersect V(D) R(288) I V(E) R(166.5) and intersectV(D) R(270) I V(E) R(171); and (19) a line extending between intersectV(D) R(270) I V(E) R(171) and intersect V(G) R(252) I V(E) R(180).Additional fragments of this pentagon complex may be defined if thiscomplex is taken into consideration in conjunction with a nose/mouthpentagon complex which is described below.

B. Pentagon Complexes with n=6

There is only one pentagon complex of the size n=6. It has a Z value of1 (See FIG. 29). It is a right iris pentagon complex 150b.

1. Lateral Iris Complex

The iris pentagon complex 150b (see FIG. 29) is positioned on the matrixpentagon complex 134b with V(B) R(144) of the iris pentagon complexcoincident with V(J) R(144) of the matrix pentagon complex and V(D) ofthe iris pentagon complex coincident with V(E) R(108) of the matrixpentagon complex. Features of the overlay system 110b are defined byfragments of this pentagon complex comprising: (1) a radial extendingfrom V(J) at 144° and extending to intersect V(J) R(144) I V(A) R(180);and (2) a radial extending from V(D) at 36° and extending to intersectV(D) R(36) I V(A) R(180).

C. Pentagon Complexes with n=5

The overlay system 110b includes two pentagon complexes of the size n=5.Both of these pentagon complexes have Z values of 1 (see FIG. 26). Oneis a lateral internal lips pentagon complex 158b and the other is alateral internal naris pentagon complex 160b. These pentagon complexesdefine various features on the lateral repose overlay system 110b.

1. Lateral Internal Lips Complex

The lateral internal lips pentagon complex 158b (FIG. 26) is positionedon the matrix pentagon complex 134b as is the internal lips complex 158of the frontal repose overlay system 110a described above. Features ofthe overlay system 110b are defined by fragments of this pentagoncomplex 158b comprising: (1) a radial extending from V(J) at 198° andextending to intersect V(J) R(198) I V(E) R(108); (2) a radial extendingfrom V(I) at 324° and extending to intersect V(I) R(324) I V(E) R(252);(3) a radial extending from V(I) at 198° and extending to intersect V(I)R(198) I V(D) R(270); and (4) a portion of a radial extending from V(D)at 270° and extending between intersect V(D) R(270) I V(I) R(198) andintersect V(D) R(270) I V(I) R(234).

A variation on the overlay system 110b is defined by fragments of thispentagon complex 158b comprising: (1) a line extending from intersectV(J) R(0/360) I V(A) R(88) and intersect V(A) R(234) I V(J) R(90); and(2) a line extending between intersect V(A) R(234) I V(J) R(90) andintersect V(A) R(216) I V(E) R(108).

2. Lateral Internal Naris Complex

The lateral internal naris pentagon complex 160b (FIG. 26) is positionedon the matrix pentagon complex 134b as is the internal naris pentagoncomplex 160 of the frontal repose overlay system 110a described above.Features of the overlay system 110b are defined by fragments of thispentagon complex 160b comprising: (1) a radial extending from V(E) at54° and extending to intersect V(E) R(54) I V(J) R(90); (2) a radialextending from V(E) at 288° and extending to intersect V(E) R(288) IV(J) R(252); (3) a radial extending from V(B) at 36° and extending tointersect V(B) R(36) I V(F) R(108); (4) a radial extending from V(B) at252° and extending to intersect V(B) R(252) I V(D) R(0/360); (5) aradial extending from V(D) at 0/360° and extending to intersect V(D)R(0/360) I V(C) R(270); (6) a radial extending from V(D) at 180° andextending to intersect V(D) R(180) I V(H) R(270); (7) a portion of aradial extending from V(J) at 90° and extending between intersect V(J)R(90) I V(A) R(234) and intersect V(J) R(90) I V(A) R(126); (8) aportion of a radial extending from V(A) at 126° and extending betweenintersect V(A) R(126) I V(F) R(270) and intersect V(A) (126) I V(F)R(216); and (9) a portion of a radial extending from V(F) at 216° andextending between intersect V(F) R(216) I V(A) R(126) and intersect V(F)R(216) I V(A) R(144).

D. Pentagon Complexes with n=4

The overlay system 110b includes four pentagon complexes of the sizen=4. These pentagon complexes have Z values of 1 (FIG. 25) and are: (1)a lateral nasal pentagon complex 170b; (2) a lateral chin buttonpentagon complex 172b; (3) a lateral eye pentagon complex 174b; and (4)a lateral teeth/lip eversion pentagon complex 224b.

1. Lateral Nasal Complex

The lateral nasal pentagon complex 170b (FIG. 25) is positioned on thematrix pentagon complex 170 as is the nasal pentagon complex of thefrontal repose overlay system 110a described above. Features of theoverlay system 110b are defined by various fragments of this pentagoncomplex 170b comprising: (1) radials connecting V(G) to V(C) and V(C) toV(H); (2) a radial extending from V(D) at 54° and extending to intersectV(D) R(54) I V(H) R(342); (3) a radial extending from V(D) at 306° andextending to intersect V(D) R(306) I V(I) R(162); (4) a radial extendingfrom V(I) at 162° and extending to intersect V(I) R(162) I V(D) R(306);(5) a radial extending from V(I) at 18° and extending to intersect V(I)R(18) I V(E) R(54); (6) a radial extending from V(I) at 36° andextending to intersect V(I) R(36) I V(E) R(108); (7) a line extendingfrom V(G) and extending to intersect V(A) R(36) I V(F) R(324); (8) aportion of a radial extending from V(E) at 108° and extending betweenintersect V(E) R(108) I V(I) R(36) and intersect, V(E) R(108) I V(I)R(72); (9) a portion of a radial extending from V(J) at 144° andextending between intersect V(J) R(144) I V(B) R(252) and intersect V(J)R(144) I V(B) R(234); (10) a portion of a radial extending from V(H) at36° and extending between intersect V(H) R(36) I V(C) R(288) andintersect V(H) R(36) I V(C) R(236); (11) a portion of a radial extendingfrom V(C) at 306° and extending between intersect V(C) R(306) I V(H)R(36) and intersect V(C) R(306) I V(H) R(18); (12) a line extendingbetween intersect V(H) R(18) I V(C) R(306) and intersect V(H) R(342) IV(D) R(54); (13) a radial extending from V(G) at 198° and extending tointersect V(G) R(198) I V(C) R(54); and (14) a radial extending fromV(C) at 54° and extending to intersect V(C) R(54) I V(G) R(198).

2. Lateral Chin Button Complex

The lateral chin button pentagon complex 172b (FIG. 25) is positioned onthe matrix pentagon complex 134b as is the chin button complex 172 ofthe frontal repose overlay system 110a described above. Features of theoverlay system 110b are defined by fragments of this pentagon complex172b comprising: (1) a portion of a radial extending from V(E) at 342°and extending between intersect V(E) R(342) I V(J) R(288) and intersectV(E) R(342) I V(J) R(234); (2) a portion of a radial extending from V(E)at 216° and extending between intersect V(E) R(216) I V(I) R(324) andintersect V(E) R(216) I V(I) R(270); (3) a portion of a radial extendingfrom V(I) at 270° and extending between intersect V(I) R(270) I V(E)R(216) and intersect V(I) R(270) I V(D) R(288); and (4) a line extendingbetween intersect V(J) R(234) I V(E) R(342) and intersect V(E) R(216) IV(I) R(270).

3. Lateral Eye Complex

The lateral eye pentagon complex 174b (FIG. 25) is positioned on thematrix pentagon complex 134b with V(B) R(180) of the eye complex 174bcoincident with V(A) R(180) of the matrix complex 134b and V(C) R(252)of the eye complex 174b coincident with V(B) R(252) of the matrixcomplex 134b. Features of the overlay system 110b are defined byfragments of this pentagon complex 174b comprising: (1) a radialconnecting V(D) to V(I); (2) a radial extending from V(D) at 54° andextending to intersect V(D) R(54) I V(C) R(306); (3) a radial extendingfrom V(I) at 324° and extending to intersect V(I) R(324) I V(E) R(216);(4) a radial extending from V(E) at 216° and extending to intersect V(E)R(216) I V(I) R(324); (5) a radial extending from V(E) at 72° andextending to intersect V(E) R(72) I V(A) R(180); (6) a portion of aradial extending from V(H) at 324° and extending between intersect V(H)R(324) I V(B) R(234) and intersect V(H) R(324) I V(I) R(36); (7) aportion of a radial extending from V(E) at 90° and extending betweenintersect V(E) R(90) I V(J) R(162) and intersect V(E) R(90) I V(J)R(144); (8) a portion of a radial extending from V(E) at 90° andextending between intersect V(E) R(90) I V(A) R(180) and intersect V(E)R(90) I V(A) R(162); (9) a portion of a radial extending from V(J) at144° and extending between intersect V(J) R(144) I V(F) R(234) andintersect V(J) R(144) I V(F) R(216); (10) a portion of a radialextending from V(F) at 216° and extending between intersect V(F) R(216)I V(A) R(180) and intersect V(F) R(216) I V(A) R(162); and (11) aportion of a radial extending from matrix complex V(G) at 288° andextending between intersect matrix complex V(G) R(288) I eye complexV(H) R(18) and intersect matrix complex V(G) R(288) I eye complex V(H)R(0/360).

4. Lateral Teeth/Lip Eversion Complex

The lateral teeth/lip eversion pentagon complex 224b (FIG. 25) ispositioned on the matrix pentagon complex 134b with V(A) R(180) of theeversion complex 224b coincident with V(A) R(180) of the matrix complex134b and V(G) R(252) of the eversion complex 224b coincident with V(G)R(252) of the matrix complex 134b. A variation of features of theoverlay system 110b are defined by fragments of this pentagon complex224b comprising: (1) a portion of a radial extending from V(A) at 198°and extending between intersect V(A) R(198) I V(E) R(126) and intersectV(A) R(198) I V(E) R(90); (2) a portion of a radial extending from V(E)at 90° and extending between intersect V(E) R(90) I V(J) R(162) andintersect V(E) R(90) I V(J) R(144); and (3) a portion of a radialextending from V(J) at 144° and extending between intersect V(J) R(144)I V(F) R(252) and intersect V(J) R(144) I V(F) R(234).

E. Pentagon Complexes with n=3

Nine pentagon complexes of the size n=3 are included in the lateralrepose overlay system 110b (See FIG. 22). These nine pentagon complexeshave Z values of 1 and include: (1) a lateral nose/mouth pentagoncomplex 184b; (2) a lateral mouth/chin pentagon complex 186b; (3) alateral chin inferior border pentagon complex 188b; (4) a lateral chinpentagon complex 190b; (5) a lateral side chin pentagon complex 192b;(6) a lateral side nose/mouth pentagon complex 208b; (7) a lateraleye/cheek pentagon complex 196b; (8) a lateral eyebrow pentagon complex200b; and (9) a lateral internal ear pentagon complex 226b.

1. Lateral Nose/Mouth Complex

The lateral nose/mouth pentagon complex 184b (FIG. 22) is positioned onthe matrix pentagon complex 134b as is the nose/mouth pentagon complex184 of the frontal repose overlay system 110a described above. Featuresof the overlay system 110b are defined by fragments of this pentagoncomplex 184b comprising: (1) a radial connecting V(J) and V(I); (2) aradial extending from V(I) at 36° and extending to intersect V(I) R(36)I V(J) R(108); (3) a radial extending from V(I) at 126° and extending tointersect V(I) R(126) I V(D) R(0/360); (4) a portion of a radialextending from V(E) at 144° and extending between intersect V(E) R(144)I V(I) R(90) and intersect V(E) R(144) I V(I) R(108); (5) a lineextending from intersect V(D) R(36) I V(H) R(324) and intersect V(D)R(54) I V(H) R(306); (6) a radial extending from V(J) at 180° andextending to intersect V(J) R(180) I V(E) R(54); (7) a line extendingbetween intersect V(A) R(234) I V(E) R(54) and intersect V(A) R(216) IV(E) R(72); (8) a portion of a radial extending along V(A) at 198° andextending between intersect V(A) R(198) I V(I) R(54) and intersect V(A)R(198) I V(E) R(90); (9) a line extending from intersect V(A) R(198) IV(E) R(90) and intersect V(J) R(126) I V(A) R(180); (10) a lineextending from intersect V(A) R(180) I V(J) R(126) and intersect V(A)R(144) I V(J) R(108); (11) a line extending from intersect V(J) R(108) IV(A) R(144) and V(A) R(126) I V(F) R(198); (12) a radial extending fromV(F) at 198° and extending to intersect V(F) R(198) I V(A) R(126); (13)a radial extending from V(F) at 336° and extending to intersect V(F)R(336) I V(J) R(18); and (14) a portion of a radial extending from V(J)at 18° and extending between intersect V(J) R(18) I V(F) R(336) andintersect nose/mouth complex V(J) R(18) I matrix complex V(B) R(270).When considering this complex in conjunction with the matrix pentagoncomplex described above, additional fragments may be describedcomprising: (15) a portion of a radial extending from matrix complexV(F) at 270° and extending between intersect matrix complex V(F) R(270)I nose/mouth complex V(F) R(351) and intersect matrix complex V(F)R(270) I matrix complex V(A) R(234).

2. Lateral Mouth/Chin Complex

The lateral mouth/chin pentagon complex 186b (FIG. 22) is positioned onthe matrix pentagon complex 134b as is the mouth/chin pentagon complex186 of the frontal repose overlay system 110a described above. Featuresof the lateral repose overlay system 110b are defined by fragments ofthis pentagon complex 186b comprising: (1) a radial extending from V(E)at 234° and extending to intersect V(E) R(234) I V(I) R(306); (2) aradial extending from V(E) at 198° and extending to intersect V(E)R(198) I V(I) R(324); (3) a radial extending from V(E) at 126° andextending to intersect V(E) R(126) I V(J) R(180); (4) a radial extendingfrom V(E) at 90° and extending to intersect V(E) R(90) I V(J) R(144) and(5) a radial extending from V(E) at 72° and extending to intersect V(E)R(72) I V(J) R(126).

3. Lateral Chin Inferior Border Complex

The lateral chin inferior border pentagon 188b (FIG. 22) complex ispositioned on the matrix pentagon complex 134b as is the chin inferiorborder pentagon complex 188 of the frontal repose overlay system 110adescribed above. Features of the lateral repose overlay system 110b aredefined by fragments of this pentagon complex 188b comprising: (1) aportion of a radial extending from V(J) at 144° and extending betweenintersect V(J) R(144) I V(A) R(234) and intersect V(J) R(144) I V(A)R(216); (2) a portion of a radial extending from V(J) at 180° andextending between intersect V(J) R(180) I V(E) R(72) and intersect V(J)R(180) I V(E) R(90); and (3) a portion of a radial extending from V(E)at 90° and extending between intersect V(E) R(90) I V(J) R(180)andintersect V(E) R(90) I V(J) R(162).

4. Lateral Chin Complex

The lateral chin pentagon complex 190b (FIG. 22) is positioned on thematrix pentagon complex 134b as is the chin pentagon complex 190 of thefrontal repose overlay system 110a described above. Features of thelateral repose overlay system 110b are defined by fragments of thispentagon complex 190b comprising: (1) a point at intersect V(J) R(198) IV(E) R(90); (2) a point at intersect V(E) R(144) I V(I) R(54); (3) apoint at intersect V(I) R(180) I V(D) R(306); and (4) a point atintersect V(I) R(216) I V(D) R(288).

5. Lateral Side Chin Complex

The lateral side chin pentagon complex 192b (FIG. 22) is positioned onthe matrix pentagon complex 134b as is the side chin pentagon complex192 of the frontal repose overlay system 110a described above. Featuresof the lateral repose overlay system 110b are defined by fragments ofthis pentagon complex 192b comprising: (1) a radial extending from V(I)at 288° and extending to intersect V(I) R(288) I V(E) R(234); (2) aradial extending from V(I) at 108° and extending to intersect V(I)R(108) I V(D) R(36); (3) a radial extending from V(A) at 72° andextending to intersect V(A) R(72) I V(B) R(342); (4) a radial extendingfrom V(A) at 90° and extending to intersect V(A) R(90) I V(F) R(0/360);(5) a radial extending from V(A) at 126° and extending to intersect V(A)R(126) I V(F) R(252); (6) a radial extending from V(A) at 198° andextending to intersect V(A) R(198) I V(F) R(252); (7) a radial extendingfrom V(A) at 234° and extending to intersect V(A) R(234) I V(J) R(144);(8) a radial extending from V(F) at 324° and extending to intersect V(F)R(324) I V(A) R(90); and (9) a radial extending from V(F) at 198° andextending to intersect V(F) R(198) I V(A) R(126). Variations of thesefeatures are defined by fragments comprising: (10) a radial extendingfrom V(A) at 72° and extending to intersect V(A) R(72) I V(F) R(18); and(11) a radial extending from V(A) at 90° and extending to intersect V(A)R(90) I V(F) R(18). When considering this complex in conjunction withthe internal lips pentagon complex described above, additional fragmentsmay be described comprising: (12) a portion of a radial extending fromV(F) at 225° and extending between intersect V(F) R(225) I V(C) R(342)and intersect V(F) R(225) I internal lips complex V(C) R(270).

6. Lateral Side Nose/Mouth Complex

The lateral side nose/mouth pentagon complex 208b (FIG. 22) ispositioned on the matrix pentagon complex 134b as is the side nose/mouthcomplex 208 of the frontal repose overlay system 110a described above.Features of the lateral repose overlay system 110b are defined byfragments of this pentagon complex 208b comprising: (1) a radialextending from V(F) at 180° and extending to intersect V(F) R(180) IV(A) R(126); (2) a radial extending from V(F) at 198° and extending tointersect V(F) R(198) I V(G) R(270); (3) a radial extending from V(G) at270° and extending to intersect V(G) R(270) I V(F) R(198); (4) a radialextending from V(B) at 72° and extending to intersect V(B) R(72) I V(F)R(126); and (5) a radial extending from V(C) at 0/360° and extending tointersect V(C) R(0/360) I V(G) R(234).

7. Lateral Eye/Cheek Complex

The lateral eye/cheek pentagon complex 196b (FIG. 22) is positioned onthe matrix pentagon complex 134b with V(I) of the eye/cheek pentagoncomplex 196b coincident with intersect matrix complex 134b V(A) R(198) Imatrix complex 134b V(C) R(306) and V(F) R(180) of the eye/cheekpentagon complex 196b coincident with matrix complex 134b V(A) R(180).Features of the overlay system 110b are defined by fragments of thispentagon complex 196b comprising: (1) a radial extending from V(H) at18° and extending to intersect V(H) R(18) I V(C) R(306); (2) a radialextending from V(F) at 18° and extending to intersect V(F) R(18) I V(B)R(342); (3) a radial extending from V(A) at 90° and extending tointersect V(A) R(96) I V(F) R(324); (4) a radial extending from V(A) at270° and extending to intersect V(A) R(270) I V(J) R(306); (5) a radialextending from V(A) to V(J); (6) a portion of a radial extending fromV(F) at 270° and extending between intersect V(F) R(270) I V(A) R(162)and intersect V(F) R(270) I V(A) R(234); (7) a portion of a radialextending from V(A) at 234° and extending between intersect V(A) R(234)I V(J) R(90) and intersect V(A) R(234) I matrix complex V(E) R(108); (8)a portion of a radial extending from matrix complex V(E) at 108° andextending between intersect matrix complex V(E) R(108) I eye/cheekcomplex V(A) R(234) and intersect matrix complex V(E) R(108) I eye/checkcomplex V(G) R(306); (9) a portion of a radial extending from V(J) at144° and extending between intersect V(J) R(144) I V(A) R(234) andintersect V(J) R(144) I V(F) R(234); (10) a portion of a radialextending from V(F) at 234° and extending between intersect V(F) R(234)I V(J) R(126) and intersect V(F) R(234) I V(J) R(162); and (11) aportion of a radial extending from V(A) at 180° and extending betweenintersect V(A) R(180) I V(J) R(126) and intersect V(A) R(180) I V(J)R(144).

8. Lateral Eve Brow Complex

The lateral eye brow pentagon complex 200b is positioned on the matrixpentagon complex 134b with V(F) R(180) of the eye brow complex 200bcoincident with V(A) R(180) of the matrix complex 134b and intersect eyebrow complex 200b V(C) R(0/360) I eye brow complex 200b V(D) R(72)coincident with intersect matrix complex 134b V(D) R(36) I matrixcomplex 134b V(E) R(108). Features of the overlay system 110b aredefined by fragments of this pentagon complex 200b comprising: (1)radials connecting V(F) to V(A) and V(A) to V(J); (2) a radial extendingfrom V(F) at 162° and extending to intersect V(F) R(162) I V(A) R(126);(3) a radial extending from V(H) at 0/360° and extending to intersectV(H) R(0/360) I V(G) R(270); (4) a portion of a radial extending alongV(H) at 324° and extending between intersect V(H) R(324) I V(I) R(108)and intersect V(H) R(324) I V(I) R(36); (5) a portion of a radialextending along V(I) at 36° and extending between intersect V(I) R(36) IV(E) R(144) and intersect V(I) R(36) I V(E) R(108); (6) a portion of aradial extending along V(I) at 54° and extending between intersect V(I)R(54) I V(D) R(0/360) and intersect V(I) R(54) I V(D) R(18); (7) aportion of a radial extending along V(A) at 144° and extending betweenintersect V(A) R(144) I V(F) R(180) and intersect V(A) R(144) I V(F)R(252); (8) a portion of a radial extending along V(F) at 252° andextending between intersect V(F) R(252) I V(A) R(144) and intersect V(F)R(252) I V(J) R(198); (9) a portion of a radial extending along V(A) at126° and extending between intersect V(A) R(126) I V(F) R(162) andintersect V(A) R(126) I V(F) R(270); (10) a portion of a radialextending along V(F) at 270° and extending between intersect V(F) R(270)I V(A) R(126) and intersect V(F) R(270) I V(A) R(198); (11) a portion ofa line extending from intersect V(A) R(180) I V(F) R(234) at 276° andextending between the intersect of this line and V(E) R(72) and theintersect of this line and V(J) R(216); (12) a portion of a radialextending from V(E) at 60° and extending between intersect V(E) R(60) IV(A) R(198) and intersect V(E) R(60) and the line described in (11)above; and (13) a portion of a radial extending from V(J) at 216° andextending to the line described in (11) above.

9. Lateral Internal Ear Complex

The lateral internal ear pentagon complex 226b is positioned on thematrix pentagon complex 134b with V(A) of the ear complex 226bcoincident with V(E) of the matrix complex 134b and V(A) R(180) of theear complex 226b coincident with V(E) R(180) of the matrix complex 134b.Features of the lateral repose overlay system 110b are defined byfragments of this pentagon complex 226b comprising: (1) radialsconnecting V(J) to V(E), V(E) to V(I), and V(I) to V(D); (2) a radialextending from V(D) at 108° and extending to intersect ear complex V(D)R(108) I matrix complex V(I) R(342); (3) a radial extending from V(H) at0/360° and extending to intersect V(H) R(0/360) I V(C) R(288); (4) aradial extending from V(H) at 180° and extending to intersect V(H)R(180) I V(C) R(234); (5) a radial extending from V(A) at 306° andextending to intersect V(A) R(306) I V(J) R(18); (6) a radial extendingfrom V(A) at 180° and extending to intersect V(A) R(180) I V(J) R(126);(7) a portion of a radial extending from V(F) at 234° and extendingbetween intersect V(F) R(234) I V(J) R(126) and intersect V(F) R(234) IV(J) R(180); (8) a portion of a radial extending from V(J) at 180° andextending between intersect V(J) R(180) I V(I) R(54) and intersect V(J)R(180) I V(I) R(36); (9) a portion of a radial extending from V(I) at36° and extending between intersect V(I) R(36) I V(E) R(108) andintersect V(I) R(36) I V(E) R(72); (10) a radial extending from V(E) at72° and extending to intersect V(E) R(72) I V(A) R(216); (11) a portionof a radial extending from V(J) at 198° and extending between intersectV(J) R(198) I V(E) R(72) and intersect V(J) R(198) I V(E) R(162); (12) aportion of a radial extending from V(E) at 162° and extending betweenintersect V(E) R(162) I V(I) R(18) and intersect V(E) R(162) I V(I)R(90); (13) a portion of a radial extending from V(I) at 90° andextending between intersect V(I) R(90) I V(E) R(162) and intersect V(I)R(90) I V(E) R(144); (14) a portion of a radial extending from V(E) at144° and extending between intersect V(E) R(144) I V(I) R(90) andintersect V(E) R(144) I V(I) R(108); (15) a portion of a radialextending from V(I) at 108° and extending between intersect V(I) R(108)I V(H) R(324) and intersect V(I) R(108) I V(H) R(342); (16) a portion ofa radial extending from V(H) at 342° and extending between intersectV(H) IR(342) I V(I) R(108) and intersect V(H) R(342) I V(I) R(90); (17)a portion of a radial extending from V(D) at 36° and extending betweenintersect V(D) R(36) I V(J) R(162) and intersect V(D) R(36) I V(J)R(144); (18) a portion of a radial extending from V(G) at 288° andextending between intersect V(G) R(288) I V(J) R(144) and intersect V(G)R(288) I V(J) R(162); (19) a portion of a radial extending from V(C) at324° and extending between intersect V(C) R(324) I V(F) R(216) andintersect V(C) R(324) I V(F) R(234); and (20) a portion of a radialextending from V(A) at 180° and between intersect V(A) R(180) I V(G)R(270) and intersect V(A) R(180) I V(G) R(297).

F. Pentagon Complexes with n=2

There are two pentagon complexes of the size n=2 (See FIG. 18) includedin the lateral repose overlay system 110b. These pentagon complexes haveZ values of 1 and are: (1) a lateral smile pentagon complex 220b; and(2) a lateral ear pentagon complex 228b.

1. Lateral Smile Complex

The lateral smile pentagon complex 220b (FIG. 18) is positioned on thematrix pentagon complex 134b as is the smile pentagon complex 220 of thefrontal repose overlay system 110a described above. Features of thelateral repose overlay system are defined by fragments of this pentagoncomplex 220b comprising: (1) a radial extending from matrix complex V(D)at 108° and extending to intersect matrix complex V(D) R(108) I smilecomplex V(I) R(162); (2) a portion of a radial extending from V(A) at216° and extending between intersect V(A) R(216) I V(J) R(108) andintersect V(A) R(216) I V(J) R(180); (3) a portion of a radial extendingfrom V(F) at 252° and extending between intersect V(F) R(252) I V(A)R(144) and intersect V(F) R(252) I V(A) R(180); (4) a point at intersectV(A) R(198) I V(R(126); and (5) a point at intersect V(D) R(36) I V(H)R(306).

2. Lateral Ear Complex

The lateral ear pentagon complex 228b is positioned on the matrixpentagon complex 134b with V(H) of the ear complex 228b coincident withV(I) of the matrix complex 134b and V(H) R(0/360) of the ear complex228b coincident with V(I) R(0/360) of the matrix complex 134b. Featuresof the lateral repose overlay system 110b are defined by fragments ofthis pentagon complex 228b comprising: (1) a portion of a radialextending from matrix complex V(D) at 342° and extending betweenintersect matrix complex V(D) R(342) I ear complex V(E) R(108) intersectmatrix complex V(D) R(342) I ear complex V(E) R(72); (2) a portion of aradial extending from matrix complex V(C) at 306° and extending betweenintersect matrix complex V(C) R(306) I ear complex V(A) R(180) intersectmatrix complex V(C) R(306) I ear complex V(A) R(234); (3) a radialextending from V(J) at 90° and extending to intersect V(J) R(90) I V(A)R(234); (4) a radial connecting V(J) and V(E); (5) a radial extendingfrom V(J) at 234° and extending to intersect V(J) R(234) I V(E) R(18);(6) a radial extending from V(E) at 18° and extending to intersect V(E)R(18) I V(J) R(234); (7) a radial extending from V(E) at 162° andextending to intersect V(E) R(162) I V(I) R(126); (8) a portion of aradial extending from V(I) at 126° and extending between intersect V(I)R(126) I V(D) R(342) and intersect V(I) R(126) I V(D) R(90) (9) aportion of a radial extending from V(D) at 90° and extending betweenintersect V(D) R(90) I V(H) R(306) and intersect V(D) R(90) I V(H)R(0/360); (10) a portion of a radial extending from V(I) at 108° andextending between intersect V(I) R(108) I V(H) R(324) and intersect V(I)R(108). I V(H) R(342); (11) a portion of a radial extending from V(H) at324° and extending between intersect V(H) R(324) I V(D) R(36) andintersect V(H) R(324) I V(D) R(0/360); (12) a portion of a radialextending from V(D) at 0/360° and extending between intersect V(D)R(0/360) I V(I) R(72) and intersect V(D) R(0/360) I V(I) R(36); (13) aportion of a radial extending from V(I) at 36° and extending betweenintersect V(I) R(36) I V(J) R(180) and intersect V(I) R(36) I V(J)R(144); (14) a portion of a radial extending from V(E) at 72° andextending between intersect V(E) R(72) I V(J) R(144) and intersect V(E)R(72) I V(J) R(126); (15) a portion of a radial extending from V(A) at198° and extending between intersect V(A) R(198) I V(J) R(126) andintersect V(A) R(198) I V(J) R(144); (16) a portion of a radialextending from V(J) at 126° and extending between intersect V(J) R(126)I V(A) R(198) and intersect V(J) R(126) I V(A) R(234); (17) a portion ofa radial extending from V(A) at 234° and extending between intersectV(A) R(234) I V(J) R(126) and intersect V(A) R(234) I V(J) R(198); and(18) a portion of a radial extending from V(J) at 198° and, extendingbetween intersect V(J) R(198) I V(E) R(54) and intersect V(J) R(198) IV(E) R(162).

III. Frontal Smiling Overlay System

A overlay system 110c comprising a frontal view of a face in a smilingconfiguration (FIG. 10A) will be described by following the sameprocedure as used above to describe overlay system 110a and overlaysystem 110b. As in the overlay systems 110a and 110b, there are sevendifferent sizes of pentagon complexes 112 involved in the overlay system110c comprising a frontal view of a face in a smiling configuration.Several features of the face in a smiling configuration are identical tothose features of the face in a repose configuration. Thus, several ofthe features which make up-the overlay system 110c have been describedabove and will not be re-described here.

A. Pentagon Complexes with n=0

The frontal smiling overlay system 110c includes only one pentagoncomplex of the size n=0. This pentagon complex has a Z value of 1 and iscalled a frontal smiling matrix pentagon complex 134c (See FIG. 16).

1. Frontal Smiling Matrix Complex

Features of the frontal smiling overlay system 110c are defined byfragments of the frontal smiling matrix pentagon complex 134a (See FIG.16) comprising: (1) a portion of a radial extending from V(F) at 180°and extending between intersect V(F) R(180) I V(A) R(126) and intersectV(F) R(180) I V(G) R(288); (2) a portion of a radial extending from V(J)at 180° and extending between intersect V(J) R(180) I V(A) R(234) andintersect V(J) R(180) I V(I) R(72); (3) a portion of a radial extendingfrom V(A) at 126° and extending between intersect V(A) R(126) I V(F)R(180) and intersect V(A) R(126) I V(F) R(270); (4) a portion of aradial extending from V(A) at 234° and extending between intersect V(A)R(234) I V(J) R(180) and intersect V(A) R(234) I V(J) R(90); (5) aportion of a radial extending from V(F) at 198° and extending betweenintersect V(F) R(198) I V(A) R(126) and intersect V(F) R(198) I V(A)R(144); (6) a portion of a radial extending from V(J) at 162° andextending between intersect V(J) R(162) I V(A) R(234) and intersect V(J)R(162) I V(A) R(216); (7) a portion of a radial extending from V(A) at144° and extending between intersect V(A) R(144) I V(F) R(198) andintersect V(A) R(144) I V(F) R(180); (8) a portion of a radial extendingfrom V(A) at 216° and extending between intersect V(A) R(216) I V(J)R(162) and intersect V(A) R(216) I V(J) R(180); (9) a portion of aradial extending from V(F) at 252° and extending between intersect V(F)R(252) I V(A) R(126) and intersect V(F) R(252) I V(A) R(180); (10) aportion of a radial extending from V(J) at 108° and extending betweenintersect V(J) R(108) I V(A) R(234) and intersect V(J) R(108) I V(A)R(180); and (11) a portion of a radial extending from V(F) at 270° andextending between intersect V(F) R(270) I V(A) R(126) and intersect V(F)R(270) I V(A) R(234).

B. Pentagon Complexes with n=6

The pentagon complexes 150c with n=6 (See FIG. 29) comprising theoverlay system 110c are identical to and describe features identical tothe iris pentagon complexes 150 described above in connection withoverlay system 110a and the features described thereby. These pentagoncomplexes 150c will not be re-described here.

C. Pentagon Complexes with n=5

There are five pentagon complexes of size n=5 in the frontal Smilingoverlay system 110c. Three of these pentagon complexes have Z values of1 (FIG, 26), one has a Z value of PHI/2 (FIG. 27), and one has a Z valueof 2/PHI (FIG. 28). The pentagon complexes with Z=1 are; (1) a smilingnasal tip complex (2) a smiling internal lips complex 158c; and (3) asmiling internal naris complex 160c. The complex with Z=PHI/2 is asmiling inner nasal tip halo complex 162c and the complex with Z=2/PHIis a smiling outer nasal tip halo complex 164c. Fragments of these fivepentagon complexes define features of the frontal smiling overlay system110c.

1. Smiling Nasal Tip Complex

The smiling nasal tip complex 156c (FIG. 26) is in the same is positionon the matrix complex 134c as is the nasal tip complex 156 of thefrontal repose overlay system 110a described above. Features of thefrontal smiling overlay system 110c are defined by fragments of thispentagon complex 156c comprising; (1) radials connecting v(F) to V(G),v(G) to V(H), V(H) to V(I), and V(I) to V(J); (2) a radial extendingfrom V(F) at 0o/360o and extending to intersect nasal tip complex V(F)R(0/360) I matrix complex V(F) R(216); (3) a radial extending from V(J)at 0o/360o and extending to intersect nasal tip complex V(J) R(0/360) Imatrix complex V(J) R(144); (4) a radial extending from V(G) at 252° andextending to intersect V(G) R(252) I V(A) R(180); (5) a radial extendingfrom V(I) at 108° and extending to intersect V(I) R(108) I V(A) R(180);(6) a radial extending from V(G) at 72° and extending to intersect nasaltip complex V(G) R(72) I matrix complex V(B) R(234); and (7) a radialextending from V(I) at 288° and extending to intersect nasal tip complexV(I) R(288) I matrix complex V(E) R(126).

2. Smiling Internal Lips Complex

The smiling internal lips complex 158c (FIG. 26) is in the same positionon the matrix complex 134a as is the internal lip complex 158 of thefrontal repose overlay system 110a described above. Features of thefrontal smiling overlay system 110a are defined by fragments of thispentagon complex 158c comprising: (1) a radial extending from V(A) at180° and extending to intersect V(A) R(180) I V(B) R(252); (2) a radialextending from intersect V(A) R(198) I V(J) R(162) to intersect V(A)R(162) I V(F) R(198); (3) a radial extending from V(B) at 198° andextending to intersect V(B) R(198) I V(G) R(342); (4) a radial extendingfrom V(B) at 0/360° and extending to intersect V(B) R(0/360) I V(A)R(108); (5) a radial extending from V(E) at 162° and extending tointersect V(E) R(162) I V(I) R(18); (6) a radial extending from V(E) at0/360° and extending to intersect V(E) R(0/360) I V(A) R(252); (7) aradial extending from intersect V(A) R(90) I V(G) R(18) through V(A) andto intersect V(A) R(270) I V(I) R(342); (8) a radial extending fromintersect V(H) R(90) I V(C) R(234) through V(H) and to intersect V(H)R(270) I V(D) R(126); (9) a portion of a radial extending from thecenter point of the complex at 270° and extending between radial V(B)R(198) and radial V(E) R(162); (10) a radial extending from V(A) at 180°and extending to the center point of the complex; (11) radial extendingfrom V(B) at 198° and extending to intersect V(B) R(198) I V(G) R(342);(12) a radial extending from V(E) at 162° and extending to intersectV(E) R(162) I V(I) R(18); (13) a radial extending from V(B) at 0/360°and extending to intersect V(B) R(0/360) I V(F) R(90); and (14) a radialextending from V(E) at 0/360° and extending to intersect V(E) R(0/360) IV(J) R(270).

3. Smiling Internal Naris Complex

The smiling internal naris complex 160c (FIG. 26) is positioned on thematrix complex 134c as is the internal naris complex 160 of the frontalrepose overlay system 110a described above. Features of the frontalsmiling overlay system 110c are defined by fragments of this pentagoncomplex 160c comprising: (1) a radial extending from V(C) to V(H) andfrom V(H) to V(D); (2) a radial extending from V(H) at 180° andextending to intersect V(H) R(180) I V(C) R(216); (3) a portion of aradial extending from V(G) at 180° and extending between intersect V(G)R(180) I V(C) R(126) and intersect V(G) R(180) I V(C) R(162); (4) aportion of a radial extending from V(I) at 180° and extending betweenintersect V(I) R(180) I V(D) R(234) and intersect V(I) R(180) I V(D)R(198); (5) a radial extending from V(G) at 252° and extending tointersect V(G) R(252) I V(A) R(180); (6) a radial extending from V(I) at108° and extending to intersect V(I) R(108) I V(A) R(180); (7) a radialextending from V(F) at 234° and extending to intersect V(F) R(234) IV(A) R(180): (8) a radial extending from V(J) at 126° and extending tointersect V(J) R(126) I V(A) R(180); (9) a portion of a radial extendingalong V(F) at 108° and extending between intersect V(F) R(108) I V(G)R(0/360) and intersect V(F) R(108) I V(G) R(36); (10) a portion of aradial extending along V(J) at 252° and extending between intersect V(J)R(252) I V(I) R(0/360) and intersect V(J) R(252) I V(I) R(324); (11) aportion of a radial extending from V(C) at 36° and extending betweenintersect V(C) R(36) I V(F) R(108) and intersect V(C) R(36) I V(F)R(72); and (12) a portion of a radial extending from V(D) at 324° andextending between intersect V(D) R(324) I V(J) R(252) and intersect V(D)R(324) I V(J) R(288).

4. Smiling Inner Nasal Tip Halo Complex

The smiling inner nasal tip halo complex 162c (FIG. 27) is positioned onthe matrix complex 134c as is the inner nasal tip halo complex 162 ofthe frontal repose overlay system 110a described above. Features of thefrontal smiling overlay system 110c are defined by fragments of thispentagon complex 162a comprising radials connecting V(B) to V(G), V(G)to V(H), V(H) to V(I), and V(I) to V(E).

5. Smiling Outer Nasal Tip Halo Complex

The smiling outer nasal tip halo complex 164c (FIG. 28) is positioned onthe matrix complex 134c as is the outer nasal tip halo complex 164 ofthe frontal repose overlay system 110a described above. Features of thefrontal smiling overlay system 110c are defined by fragments of thispentagon complex 164c comprising: (1) radials connecting V(F) to V(G),and V(I) to V(J); (2) a radial extending from V(F) at 0/360° andextending to intersect nasal tip halo complex V(F) R(0/360) I matrixcomplex, V(B) R(252); (3) a radial extending from V(J) at 0/360° andextending to intersect nasal tip halo complex V(J) R(0/360) I matrixcomplex V(E) R(108); (4) a radial extending from V(G) at 234° andextending to intersect V(G) R(234) I matrix complex V(G) R(270); and (5)a radial extending from V(I) at 126° and extending to intersect V(I)R(126) I matrix complex (I) R(90).

D. Pentagon Complexes with n=4

There are five pentagon complexes of size n=4 (see FIG. 25) in thefrontal smiling overlay system 110c. All five of these pentagoncomplexes have Z values of 1 and are: (1) a smiling nasal complex 170c;(2) a chin button complex 172c; (3) a smiling right eye complex 174c;(4) a smiling left eye complex 176c; and (5) a smiling teeth/lipeversion complex 224c. Fragments of these five pentagon complexes definevarious features of the frontal smiling overlay system 110c.

1. Smiling Nasal Complex

The smiling nasal complex 170c (FIG. 25) is positioned on the matrixcomplex 134c as is the nasal complex 170 of the frontal repose overlaysystem 110a described above. Features of the frontal smiling overlaysystem 110c are defined by fragments of this pentagon complex 170ccomprising: (1) a radial extending from V(B) at 144° and extending tointersect V(B) R(144) I V(G) R(54); (2) a radial extending from V(E) at216° and extending to intersect V(E) R(216) I V(I) R(306); (3) a radialextending from V(B) at 288° and extending to intersect V(B) R(288) IV(F) R(162); (4) a radial extending from V(E) at 72° and extending tointersect V(E) R(72) I V(J) R(198); (5) a line extending betweenintersect V(B) R(288) I V(F) R(162) and intersect V(B) R(270) I V(F)R(180); and (6) a line extending between intersect V(E) R(72) I V(J)R(198) and intersect V(E) R(90) I V(J) R(180).

2. Smiling Chin Button Complex

The smiling chin button complex 172c (FIG. 25) is positioned on thematrix complex 134c as is the chin button complex 172 of the frontalrepose overlay system 110a described above. Features of the frontalsmiling overlay system 110c are defined by fragments of this pentagoncomplex 172c comprising: (1) a radial connecting V(G) and V(I); (2) aradial connecting V(B) and V(E); (3) a radial extending from V(G) at 54°and extending to intersect V(G) R(54) I V(B) R(72); (4) a radialextending from V(I) at 306° and extending to intersect V(I) R(306) IV(E) R(288); (5) a radial extending from V(B) at 18° and extending tointersect V(B) R(18) I V(F) R(36); and (6) a radial extending from V(E)at 342° and extending to intersect V(E) R(342) I V(J) R(324).

3. Smiling Right Eye Complex

The smiling right eye complex 174c (FIG. 25) is positioned on the matrixcomplex 134c as is the right eye complex 174 of the frontal reposeoverlay system 110a described above. In addition, the fragments of thesmiling right eye complex 174c which define features of the overlaysystem 110c are identical to the fragments of the right eye complex 174which define features of the overlay system 110a described above.

4. Smiling Left Eye Complex

The smiling left eye complex 176c (FIG. 25) is positioned on the matrixcomplex 134c as is the left eye complex 176 of the frontal reposeoverlay system 110a described above. In addition, the fragments of thesmiling left eye complex 176c which define features of the overlaysystem 110c are identical to the fragments of the left eye complex 176which define features of the overlay system 110a described above.

5. Smiling Teeth/Lip Eversion Complex

The smiling teeth/lip eversion pentagon complex 224° (FIG. 25) ispositioned on the matrix pentagon complex 134a with V(A) of the eversioncomplex 224c coincident with intersect matrix complex 134c V(B) R(234) Imatrix complex 134c V(E) R(126) and V(G) of the eversion complex 224ccoincident with intersect. matrix complex 134a V(D) R(72) I matrixcomplex 134a V(F) R(198). Features of the overlay system 110c aredefined by fragments of this pentagon complex 224c comprising: (1) aradial connecting V(B) and V(E); (2) a radial extending from V(F) at252° and extending to intersect V(F) R(252) I V(A) R(180); (3) a radialextending from V(J) at 108° and extending to intersect V(J) R(108) IV(A) R(180); (4) a radial extending from V(C) at 54° and extending tointersect V(C) R(54) I V(G) R(72); (5) a radial extending from V(D) at306° and extending to intersect V(D) R(306) I V(I) R(288); (6) a portionof a radial extending from V(G) at 270° and extending between intersectV(G): R(270) I V(B) R(198) and intersect V(G) R(270) I V(E) R(162); (7)a portion of a radial extending from V(A) at 144° and extending betweenintersect V(A) R(144) I V(B) R(270) and intersect V(A) R(144) I V(B)R(252); (8) a portion of a radial extending from V(A) at 216° andextending between intersect V(A) R(216) I V(E) R(90) and intersect V(A)R(216) I V(E) R(108); (9) a portion of a radial extending from V(F) at180° and extending between intersect V(F) R(180) I V(B) R(252) andintersect V(F) R(180) I V(J) R(126); (10) a portion of a radialextending from V(J) at 180° and extending between intersect V(J) R(180)IV(E) R(108) and intersect V(J) R(180) I V(F) R(234); and (11) a radialbetween V(C) and V(D).

Two additional features of the overlay system 110a are defined byparallel fragments of this pentagon complex 224a comprising: (12) twofragments that are parallel to the fragments described in (2) and (3)above and that extend between intersect V(B) R(270) I V(F) R(216) andintersect V(E) R(90) I V(J) R(144); and (13) two fragments that areparallel to the fragments described in (2) and (3) above and that extendbetween intersect V(F) R(198) I V(G) R(306) and intersect V(J) R(162) IV(I) R(54).

There are three additional fragments of the eversion pentagon complex224c which define features of the overlay system 110c, however, asmiling nose/mouth complex 184c must also be considered to fullydescribe these additional fragments. The smiling nose/mouth complex 184cis described below and, therefore, these additional fragments of theeversion pentagon complex 224c will also be described below.

E. Pentagon Complexes with n=3

There are sixteen pentagon complexes of size n=3 in the frontal smilingoverlay system 110c. Fourteen of these pentagon complexes have Z valuesof 1 (see FIG. 22) and two have Z values of (Φ)^(1/3). (See FIG. 22) Thepentagon complexes with Z=1 include: (1) a smiling nose/mouth complex184c; (2) a smiling mouth/chin complex 186c; (3) a smiling chin inferiorborder complex 188c; (4) a smiling chin complex 190c; (5) a smilingright side chin complex 192c; (6) a smiling left side chin complex 194c;(7) a smiling right eye/cheek complex :196c; (8) a smiling lefteye/cheek complex 198c; (9) a smiling right eye brow complex 200c; (10)a smiling left eye brow complex 202a; (11) a smiling right cheek complex204c; (12) a smiling left cheek-complex 206c; (13) a smiling rightnose/mouth complex 208c; and (14) a smiling left nose/mouth complex210c. The two complexes with Z=(Φ)^(1/3) include: (15) a smiling righteye brow/cheek complex 212c; and (16) a smiling left eye brow/cheekcomplex 214c. Fragments of these sixteen pentagon complexes definevarious features of the frontal smiling overlay system 110c of thepresent invention.

1. Smiling Nose/Mouth Complex

The smiling nose/mouth complex 184c (FIG. 22) of the frontal smilingoverlay system 110c is positioned on the matrix complex 134a as is thenose/mouth complex 184 of the frontal repose overlay system 110adescribed above. Features of the frontal smiling overlay system 110c aredefined by fragments of this pentagon complex 184c comprising: (1) aradial connecting V(B) and V(E); (2) a portion of a radial Extendingfrom V(A) at 180° and extending between intersect V(A) R(180) I V(B)R(252) and intersect V(A) R(180) I V(B) R(234); (3) a portion of aradial extending from V(F) at 180° and extending between intersect V(F)R(180) I V(G) R(288) and intersect V(F) R(180) I V(G) R(306); (4) aportion of a radial extending from V(J) at 180° and extending betweenintersect V(J) R(180) I V(I) R(54) and intersect V(J) R(180) I V(I)R(72); (5) a portion of a radial extending from V(H) at 36° andextending between intersect V(H) R(36) I V(G) R(270) and intersect V(H)R(36) I V(G) R(288); (6) a portion of a radial extending from V(H) at324° and extending between intersect V(H) R(324) I V(I) R(90) andintersect V(H) R(324) I V(I) R(72); (7) a portion of a radial extendingfrom V(F) at 162° and extending between intersect V(F) R(162) I V(B)R(198) and intersect V(F) R(162) I V(B) R(216); (8) a portion of aradial extending from V(J) at 198° and extending between intersect V(J)R(198) I V(E) R(162) and intersect V(J) R(198) I V(E) R(144); (9) aportion of a radial extending from V(B) at 198° and extending betweenintersect V(B) R(198) I V(G) R(342) and intersect V(B) R(198) I V(G)R(306); (10) a portion of a radial extending from V(E) at 162° andextending between intersect V(E) R(162) I V(I) R(18) and intersect V(E)R(162) I V(I) R(54); (11) a line connecting intersect V(B) R(198) I V(G)R(342) and intersect V(E) R(162) I V(I) R(18); (12) a line connectingintersect V(B) R(216) I V(G) R(342) and intersect V(B) R(252) I V(G)R(288); (13) a line connecting intersect V(E) R(144) I V(I) R(18) andintersect V(E) R(108) I V(I) R(72); (14) a radial extending from V(C)and extending to intersect V(B) R(216) I V(G) R(324); (15) a portion ofthe radial described in (14) above extending between its intersect withV(G) R(288) and its intersect with V(G) R(324); (16) a radial extendingfrom V(D) and extending to intersect V(E) R(144) I V(I) R(36); (17) aportion of the radial described in (16) above extending between itsintersect with V(I) R(72) and its intersect with V(I) R(36); (18) aportion of a radial extending from V(G) at 288° and extending betweenintersect V(G) R(288) I V(F) R(180) and intersect V(G) R(288) and theradial described-in (14 ) above; (19) a portion of a radial extendingfrom V(I) at 72° and extending between intersect V(I) R(72) I V(J)R(180) and intersect V(I) R(72) and the radial described in (16) above;(20) a portion of a radial extending from V(G) at 306° and extendingbetween intersect V(G) R(306) I V(B) R(198) and intersect V(G) R(306)and the radial described in (14) above; and (21) a portion of a radialextending from V(I) at 54° and extending between intersect V(I) R(54) IV(E) R(162) and intersect V(I) R(54) and the radial described in (16)above.

When considering this complex 184c in conjunction with the internal lipspentagon complex 158c described above, additional fragments may bedescribed comprising: (22) a portion of a radial extending from V(G) at270° and extending between intersect V(G) R(270) I V(B) R(216) andintersect V(G) R(270) I internal lips complex V(B) R(198); and (23) aportion of a radial extending from V(I) at 90° and extending betweenintersect V(I) R(90) I V(E) R(144) and intersect V(I) R(90) I internallips complex V(E) R(162).

2. Smiling Mouth/Chin Complex

The smiling mouth/chin complex 186c (FIG. 22) of the frontal smilingoverlay system 110c is positioned on the matrix complex 134c as is themouth/chin complex 186c of the frontal repose overlay system 110adescribed above. Features of the frontal smiling overlay system 110c aredefined by fragments of this pentagon complex 186c comprising: (1)radials connecting V(B) to V(C), V(C) to V(D), and V(D) to V(E); (2) aradial extending from V(B) at 234° and extending to intersect V(B)R(234) I V(A) R(162); and (3) a radial extending from V(E) at 126° andextending to intersect V(E) R(126) I V(A) R(198).

3. Smiling Chin Inferior Border Complex

The smiling chin inferior border complex 188c (FIG. 22) of the frontalsmiling overlay system 110c is positioned on the matrix complex 134c asis the chin inferior border complex 188 of the frontal repose overlaysystem 110a described above. Features of the frontal smiling overlaysystem 110c are defined by fragments of this pentagon complex 188ccomprising: (1) a radial connecting V(C) to V(D); (2) a radial extendingfrom V(C) at 54° and extending to intersect V(C) R(54) I V(B) R(126);(3) a radial extending from V(D) at 306° and extending to intersect V(D)R(306) I V(E) R(234); and (4) a line extending between intersect V(B)R(234) I V(G) R(324) and intersect V(E) R(126) I V(I) R(36).

4. Smiling Chin Complex

The smiling chin complex 190c (FIG. 22) of the frontal smiling overlaysystem 110c is positioned on the matrix complex 134c as is the chincomplex 190 of the frontal repose overlay system 110a described above.Features of the frontal smiling overlay system 110c are defined byfragments of this pentagon complex 190c comprising: (1) a radialextending from V(G) at 234° and extending to intersect V(G) R(234) IV(C) R(0/360); and (2) a radial extending from V(I) at 126° andextending to intersect V(I) R(126) I V(C) R(0/360).

5. Smiling Right Side Chin Complex

The smiling right side chin complex 192c of the frontal smiling overlaysystem 110c is positioned on the matrix complex 134c as is the rightside chin complex 192 of the frontal repose overlay system 110adescribed above. Features of the frontal smiling overlay system 110c aredefined by fragments of this pentagon complex 192a comprising: (1) aradial extending from V(A) at 126° and extending to intersect V(A)R(126) I V(F) R(180); and (2) a radial extending from V(A) at 162° andextending to intersect V(A) R(162) I V(B) R(234).

6. Smiling Left Side Chin Complex

The smiling left side chin complex 194c (FIG. 22) of the frontal smilingoverlay system 110c is positioned on the matrix complex 134a as is theleft side chin complex 196 of the frontal repose overlay system 110adescribed above. Features of the frontal smiling overlay system 110c aredefined by fragments of this pentagon complex 196c comprising: (1) aradial extending from V(A) at 234° and extending to intersect V(A)R(234) I V(J) R(180); and (2) a radial extending from V(A) at 198° andextending to intersect V(A) R(198) I V(E) R(126).

7. Smiling Right Eye/Cheek Complex

The smiling right eye/cheek complex of the frontal smiling overlaysystem 110c is positioned on the matrix complex as is the righteye/cheek complex of the frontal repose overlay system 110a describedabove. Features of the frontal smiling overlay system 110c are definedby fragments of this pentagon complex comprising: (1) radials connectingV(B) to V(C), V(C) to V(H), V(H) to V(D), V(D) to V(I), and V(I) toV(E); (2) radials connecting V(J). to V(A) and V(A) to V(F); (3) aradial connecting V(D) to V(E); (4) a radial extending from V(E) at 54°and extending to intersect V(E) R(54) I V(J) R(198); (5) a radialextending from V(B) at 216° and extending to intersect V(B) R(216) IV(C) R(0/360); (6) a radial extending from V(B) at 234° and extending tointersect V(B) R(234) I V(C) R(342); (7) a radial extending from V(C) at0/360° and extending to intersect V(C) R(0/360) I V(B) R(216); (8) aradial extending from V(C) at 342° and extending to intersect V(C)R(342) I V(B) R(234); (9) a portion of a radial extending from V(B) at252° and extending between intersect V(B) R(252) I V(F) R(198) andintersect V(B) R(252) I V(A) R(180); (10) a portion of a radialextending from V(E) at 108° and extending between intersect V(E) R(108)I V(J) R(162) and intersect V(E) R(108) I V(A) R(180); (11) a portion ofa radial extending from V(B) at 270° and extending between intersectV(B) R(270) I V(F) R(180) and intersect V(B) R(270) I V(F) R(162); (12)a portion of a radial extending from V(F) at 162° and extending betweenintersect V(F) R(162) I V(B) R(270) and intersect V(F) R(162) I V(B)R(306); (13) a portion of a radial extending from V(B) at 306° andextending between intersect V(B) R(306) I V(F) R(162) and intersect V(B)R(306) I V(F) R(270); (14) a portion of a radial extending from V(F) at270° and extending between intersect V(F) R(270) I V(B) R(306) andintersect V(F) R(270) I V(A) R(234); (15) a portion of a radialextending from V(A) at 234° and extending between intersect V(A) R(234)I V(F) R(270) and intersect V(A) R(234) I V(J) R(144); (16) a portion ofa radial extending from V(F) at 180° and extending between intersectV(F) R(180) I V(B) R(306) and intersect V(F) R(180) I V(B) R(288); (17)a portion of a radial extending from V(J) at 144° and extending betweenintersect V(J) R(144) I V(F) R(252) and intersect eye/cheek complex V(J)R(144) I matrix complex V(F) R(234); (18) a portion of a radialextending from matrix complex V(F) at 234° and extending betweenintersect matrix complex V(F) R(234) I eye/cheek complex V(J) R(144) andintersect matrix complex V(F) R(234) I eye/cheek complex V(J) R(126);and (19) a radial extending from V(C) at 162° and extending to intersectV(C) R(162) I matrix complex V(G) R(270).

8. Smiling Left Eve/Cheek Complex

The smiling left eye/cheek complex 198c (FIG. 22) of the frontal smilingoverlay system 110c is positioned on the matrix complex 134c as is theleft eye/cheek complex 198 of the frontal repose overlay system 110adescribed above. Features of the frontal smiling overlay system 110c aredefined by fragments of this pentagon complex 198c comprising: (1)radials connecting V(B) to V(G), V(G) to V(C), V(C) to V(H), V(H) toV(D), and V(D) to V(E); (2) radials connecting V(J) to V(A) and V(A) toV(F); (3) a radial connecting V(B) to V(C); (4) a radial extending fromV(B) at 306° and extending to intersect V(B) R(306) I V(F) R(162); (5) aradial extending from V(E) at 144° and extending to intersect V(E)R(144) I V(D) R(0/360); (6) a radial extending from V(D) at 0/360° andextending to intersect V(D) R(0/360) I V(E) R(144); (7) a radialextending from V(E) at 126° and extending to intersect V(E) R(126) IV(D) R(18); (8) a radial extending from V(D) at 18° and extending tointersect V(D) R(18) I V(E) R(126); (9) a portion of a radial extendingfrom V(E) at 108° and extending between intersect V(E) R(108) I V(J)R(162) and intersect V(E) R(108) I V(A) R(180); (10) a portion of aradial extending from V(B) at 252° and extending between intersect V(B)R(252) I V(F) R(198) and intersect V(B) R(252) I V(A) R(180); (11) aportion of a radial extending from V(B) at 306° and extending betweenintersect V(B) R(306) I V(F) R(216) and intersect V(B) R(306) I V(F)R(270); (12) a portion of a radial extending from V(F) at 270° andextending between intersect V(F) R(270) I V(A) R(126) and intersect V(F)R(270) I V(A) R(234); (13) a portion of a radial extending from V(A) at234° and extending between intersect V(A) R(234) I V(J) R(90) andintersect V(A) R(234) I V(J) R(198); (14) a portion of a radialextending from V(J) at 198° and extending between intersect V(J) R(198)I V(E) R(54) and intersect V(J) R(198) I V(E) R(90); (15) a portion of aradial extending from V(E) at 90° and extending between intersect V(E)R(90) I V(J) R(198) and intersect V(E) R(90) I V(J) R(180); (16) aportion of a radial extending from V(J) at 180° and extending betweenintersect V(J) R(180) I V(E) R(54) and intersect V(J) R(180) I V(E)R(72); (17) a portion of a radial extending from V(F) at 216° andextending between intersect V(F) R(216) I V(J) R(108) and intersecteye/cheek complex V(F) R(216) I matrix complex V(J) R(126); (18) aportion of a radial extending from matrix complex V(J) at 126° andextending between intersect matrix complex V(J) R(126) I eye/cheekcomplex V(F) R(216) and intersect matrix complex V(J) R(126) I eye/cheekcomplex V(F) R(234); and (19) a radial extending from V(D) at 198° andextending to intersect V(D) R(198) I matrix complex V(I) R(90).

9. Smiling Right Eye Brow Complex I

The smiling right eye brow complex 200c (FIG. 22) of the frontal smilingoverlay system 110c is positioned on the matrix complex 134c as is theright eye brow complex 200 of the frontal repose overlay system 110adescribed above. Features of the frontal smiling overlay system 110c aredefined by fragments of this pentagon complex 200c comprising: (1) aradial connecting V(G) to V(C); (2) a radial extending from V(G) at 198°and extending to intersect V(G) R(198) I V(C) R(180); (3) a radialextending from V(G) at 234° and extending to intersect V(G) R(234) IV(C) R(288); (4) a radial extending from V(C) at 180° and extending tointersect V(C) R(180) I V(G) R(198); (5) a radial extending from V(I) at162° and extending to intersect V(I) R(162) I V(H) R(234); (6) a radialextending from V(I) at 180° and extending to intersect V(I) R(180) IV(D) R(252); (7) a radial extending from V(I) at 54° and extending tointersect V(I) R(54) I V(E) R(144); (8) a radial extending from V(J) at234° and extending to intersect V(J) R(234) I V(E) R(288); (9) a radialextending from V(J) at 108° and extending to intersect V(J) R(108) IV(F) R(162); (10) a radial extending from eye brow complex V(A) at 108°and extending to intersect eye brow complex V(A) R(108) I matrix complexV(A) R(198); (11) a radial extending from eye brow complex V(A) at 288°and extending to intersect eye brow complex V(A) R(288) I matrix complexV(A) R(216); (12) a portion of a radial extending from V(G) at 306° andextending between intersect V(G) R(306) I V(A) R(162) and intersect V(G)R(306) I V(B) R(216); (13) a portion of a radial extending from V(G) at270° and extending between intersect V(G) R(270) I V(C). R(18) andintersect V(G) R(270) I V(C) R(0/360); (14) a portion of a radialextending from V(G) at 252° and extending between intersect V(G) R(252)I V(H) R(36) and intersect V(G) R(252) I V(H) R(0/360); (15) a portionof a radial extending from V(I) at 108° and extending between intersectV(I) R(108) I V(H) R(324) and intersect V(I) R(108) I V(H) R(0/360); and(16) a portion of a radial extending from V(I) at 54° and extendingbetween intersect V(I) R(54) I V(E) R(126) and intersect V(I) R(54) IV(A) R(198).

10. Smiling Left Eye Brow Complex

The smiling left eye brow complex 202c (FIG. 22) of the frontal smilingoverlay system 110c is positioned on the matrix complex 134a as is theleft eye brow complex 202 of the frontal repose overlay system 110adescribed above. Features of the frontal smiling overlay system 110c aredefined by different fragments of this pentagon complex 202c comprising:(1) a radial connecting V(I) to V(D); (2) a radial extending from V(I)at 162° and extending to intersect V(I) R(162) I V(D) R(180); (3) aradial extending from V(I) at 126° and extending to intersect V(I)R(126) I V(D) R(72); (4) a radial extending from V(D) at 180° andextending to intersect V(D) R(180) I V(I) R(162); (5) a radial extendingfrom V(G) at 198° and extending to intersect V(G) R(198) I V(H) R(126);(6) a radial extending from V(G) at 180° and extending to intersect V(G)R(180) I V(C) R(108); (7) a radial extending from V(G) at 306° andextending to intersect V(G) R(306) I V(B) R(216); (8) a radial extendingfrom V(F) at 126° and extending to intersect V(F) R(126) I V(B) R(72);(9) a radial extending from V(F) at 252° and extending to intersect V(F)R(252) I V(J) R(198); (10) a radial extending from eye brow complex V(A)at 252° and extending to intersect eye brow complex V(A) R(252) I matrixcomplex V(A) R(162); (11) a radial extending from eye brow complex V(A)at 72° and extending to intersect eye brow complex V(A) R(72) I matrixcomplex V(A) R(144); (12) a portion of a radial extending from V(I) at54° and extending between intersect V(I) R(54) I V(A) R(198) andintersect V(I) R(54) I V(E) R(144); (13) a portion of a radial extendingfrom V(I) at 90° and extending between intersect V(I) R(90) I V(D)R(342) and intersect V(I) R(90) I V(D) R(0/360); (14) a portion of aradial extending from V(I) at 108° and extending between intersect V(I)R(108) I V(H) R(324) and intersect V(I) R(108) I V(H) R(0/360); (15) aportion of a radial extending from V(G) at 252° and extending betweenintersect V(G) R(252) I V(H) R(36) and intersect V(G) R(252) I V(H)R(0/360); and (16) a portion of a radial extending from V(G) at 306° andextending between intersect V(G) R(306) I V(B) R(234) and intersect V(G)R(306) I V(A) R(162).

11. Smiling Right Cheek Complex

The smiling right cheek complex 204c (FIG. 22) of the frontal smilingoverlay system 110a is positioned on the matrix complex 134a as is theright cheek complex 204 of the frontal repose overlay system 110adescribed above. Features of the frontal smiling overlay system 110c aredefined by fragments of this pentagon complex 204c comprising: (1) aradial connecting V(D) to V(E); (2) a radial extending from V(D) at 144°and extending to intersect V(D) R(144) I V(H) R(180); (3) a radialextending from V(C) at 324° and extending to intersect V(C) R(324) IV(G) R(270); (4) a radial extending from V(G) at 270° and extending tointersect V(G) R(270) I V(C) R(324); and (5) a point at intersect V(A)R(198) I V(B) R(252).

12. Smiling Left Cheek Complex

The smiling left cheek complex 206c (FIG. 22) of the frontal smilingoverlay system 110c is positioned on the matrix complex 134c as is theleft cheek complex 206 of the frontal repose overlay system 110adescribed above. Features of the frontal smiling overlay system 110c aredefined by fragments of this pentagon complex 206c comprising: (1) aradial connecting V(B) to V(C); (2) a radial extending from V(C) at 216°and extending to intersect V(C) R(216) I V(H) R(180); (3) a radialextending from V(D) at 36° and extending to intersect V(D) R(36) I V(I)R(90); (4) a radial extending from V(I) at 90° and extending tointersect V(I) R(90) I V(D) R(36); and (5) a point at intersect V(A)R(162) I V(E) R(108).

13. Smiling Right Nose/Mouth Complex

The smiling right nose/mouth complex 208 (FIG. 22) of the frontalsmiling overlay system 110c is in the same position on the matrixcomplex 134c as is the right nose/mouth complex 208 of the frontalrepose overlay system 110a described above. Features of the frontalsmiling overlay system 110c are defined by fragments of this pentagoncomplex 208c comprising: (1) a radial connecting V(D) to V(I); (2) aradial extending from V(D) at 144° and extending to intersect V(D)R(144) I V(H) R(234); (3) a radial extending from V(I) at 342° andextending to intersect V(I) R(342) I V(J) R(270); (4) a radial extendingfrom V(H) at 162° and extending to intersect V(H) R(162) I V(D) R(126);(5) a radial extending from V(H) at 342° and extending to intersect V(H)R(342) I V(E) R(90); (6) a radial extending from V(F) at 126° andextending to intersect V(F) R(126) I V(B) R(72); (7) a portion of aradial extending from nose mouth complex V(C) at 324° and extendingbetween intersect nose/mouth complex V(C) R(324) I nose/mouth complexV(D) R(54) and intersect nose/mouth complex V(C) R(324) I matrix complexV(D) R(36); (8) a portion of a radial extending from V(C) at 342° andextending between intersect V(C) R(342) I V(G) R(288) and intersect V(C)R(342) I V(F) R(198); (9) a portion of a radial extending from V(F) at198° and extending between intersect V(F) R(198) I V(G) R(306) andintersect V(F) R(198) I V(C) R(342); (10) a portion of a radialextending from V(C) at 0/360° and extending between intersect V(C)R(0/360) I V(G) R(288) and intersect V(C) R(0/360) I V(G) R(306); and(11) a portion of a radial extending from V(B) at 198° and extendingbetween intersect V(B) R(198) I V(G) R(342) and intersect V(B) R(198) IV(G) R(324).

14. Smiling Left Nose/Mouth Complex

The smiling left nose/mouth complex 210c (FIG. 22) of the frontalsmiling overlay system 110c is positioned on the matrix complex 134c asis the left nose/mouth complex 210 of the frontal repose overlay system110a described above. Features of the frontal smiling overlay system110c are defined by fragments of this pentagon complex 210c comprising:(1) a radial connecting V(C) to V(G); (2) a radial extending from V(C)at 216° and extending to intersect V(C) R(216) I V(H) R(126); (3) aradial extending from V(G) at 18° and extending to intersect V(G) R(18)I V(F) R(90); (4) a radial extending from V(H) at 198° and extending tointersect V(H) R(198) I V(C) R(234); (5) a radial extending from V(H) at18° and extending to intersect V(H) R(18) I V(B) R(270); (6) a radialextending from V(J) at 234° and extending to intersect V(J) R(234) IV(E) R(288); (7) a portion of a radial extending from nose/mouth complexV(D) at 36° and extending between intersect nose/mouth complex V(D)R(36) I nose/mouth complex V(C) R(306) and intersect nose/mouth V(D)R(36) I matrix complex V(C) R(324); (8) a portion of a radialextending-from V(D) at 18° and extending between intersect V(D) R(18) IV(I) R(72) and intersect V(D) R(18) I V(J) R(162); (9) a portion of aradial extending from V(J) at 162° and extending between intersect V(J)R(162) I V(I) R(54) and intersect V(J) R(162) I V(D) R(18); (10) aportion of a radial extending from V(D) at 0/360° and extending betweenintersect V(D) R(0/360) I V(I) R(72) and intersect V(D) R(0/360) I V(I)R(54); and (11) a portion of a radial extending from V(E) at 162° andextending between intersect V(E) R(162) I V(I) R(18) and intersect V(E)R(162) I V(I) R(36).

15. Smiling Right Eye Brow/Cheek Complex

The smiling right eye brow/cheek complex 212c (FIG. 23) of the frontalsmiling overlay system 110c is positioned on the matrix complex 134c asis the right eye brow/cheek complex 212 of the frontal repose overlaysystem 110a described above. Features of the frontal smiling overlaysystem 110c are defined by fragments of this pentagon complex 212ccomprising: (1) radials connecting V(H) to V(D), V(D) to V(I), V(I) toV(E), V(E) to V(J), and V(J) to V(F); (2) a radial connecting V(B) toV(G); (3) a radial extending from V(B) at 216° and extending tointersect V(B) R(216) I V(G) R(288); (4) a radial extending from V(B) at198° and extending to intersect V(B) R(198) I V(G) R(324); (5) a radialextending from V(G) at 180° and extending to intersect V(G) R(180) IV(C) R(72); (6) a radial extending from V(H) at 306° and extending tointersect V(H) R(306) I V(D) R(18); (7) a radial extending from V(H) at324° and extending to intersect V(H) R(324) I V(D) R(0/360); (8) aradial extending from eye brow/cheek complex V(D) at 144° and extendingto intersect eye brow/cheek complex V(D) R(144) I matrix complex V(E)R(144); (9) a radial extending from V(E) at 126° and extending tointersect V(E) R(126) I V(I) R(72); (10) a radial extending from V(E) at108° and extending to intersect V(E) R(108) I V(A) R(198); (11) a radialextending from V(E) at 72° and extending to intersect V(E) R(72) I V(A)R(180); (12) a radial extending from V(E) at 54° and extending tointersect V(E) R(54) I V(J) R(144); (13) a portion of a radial extendingfrom V(B) at 234° and extending between intersect V(B) R(234) I V(J)R(126) and intersect V(B) R(234) I matrix complex V(J) R(162); (14) aportion of a radial extending from V(J) at 108° and extending betweenintersect, V(J) R(108) I V(F) R(252) and intersect V(J) R(108) I V(F)R(162); and (15) a portion of a radial extending from V(B) at 252° andextending between intersect V(B) R(252) I V(E) R(117) and intersect V(B)R(252) I V(E) R(126).

16. Smiling Left Eve Brow/Cheek Complex

The smiling left eye brow/cheek complex 214c (FIG. 23) of the frontalsmiling overlay system 110c is positioned on the matrix complex 134c asis the left eye brow/cheek complex 214 of the frontal repose overlaysystem 110a described above. Features of the frontal smiling overlaysystem 110c are defined by fragments of this pentagon complex 214ccomprising: (1) radials connecting V(C) to V(G), V(G) to V(B), V(B) toV(F), V(F) to V(A), and V(A) to V(E); (2) a radial connecting V(D) toV(I); (3) a radial extending from V(D) at 144° and extending tointersect V(D) R(144) I V(H) R(252); (4) a radial extending from V(I) at126° and extending to intersect V(I) R(126) I V(D) R(18); (5) a radialextending from V(I) at 108° and extending to intersect V(I) R(108) IV(D) R(36); (6) a radial extending from V(G) at 180° and extending tointersect V(G) R(180) I V(C) R(72); (7) a radial extending from V(C) at18° and extending to intersect V(C) R(18) I V(G) R(270); (8) a radialextending from V(C) at 0/360° and extending to intersect V(C) R(0/360) IV(B) R(252); (9) a radial extending from V(F) at 198° and extending tointersect V(F) R(198) I V(B) R(252); (10) a radial extending from V(F)at 216° and extending to intersect V(F) R(216) I V(G) R(306); (11) aradial extending from V(F) at 252° and extending to intersect V(F)R(252) I V(J) R(144); (12) a radial extending from V(F) at 270° andextending to intersect V(F) R(270) I V(A) R(198); (13) a portion of aradial extending from V(I) at 90° and extending between intersect V(I)R(90) I V(A) R(198) and intersect V(I). R(90) I V(E) R(126); (14) aportion of a radial extending from V(A) at 216° and extending betweenintersect V(A) R(216) I V(E) R(72) and intersect V(A) R(216) I V(E)R(162); and (15) a portion of a radial extending from V(B) at 252° andextending between intersect V(B) R(252) I V(F) R(207) and intersect V(B)R(252) I V(F) R(207).

E. Pentagon Complexes with n=2

There are four pentagon complexes of size n=2 in the frontal smilingoverlay system 110c. One of these pentagon complexes has a Z value of 1(see FIG. 18) and is a smiling smile pentagon complex 220c, one has a Zvalue of (2/Φ)×(Φ^(1/3)) (see FIG. 21) and is a smiling external dimplepentagon complex 230c, one has a Z value of 2/Φ (FIG. 20) and is asmiling dimple pentagon complex 232c, and the last one has a Z value ofΦ/2 (FIG. 19) and is a smiling internal smile pentagon complex 234c.Fragments of these four complexes define various features of the frontalsmiling overlay system 110c of the present invention.

1. Smiling Smile Complex

The smiling smile pentagon complex 220c (FIG. 18) is positioned on thematrix pentagon complex 134a with V(A) R(180) of the smile complex 220ccoincident with V(A) R(180) of the matrix complex 134c and V(F) R(234)of the smile complex 220c coincident with V(B) R(234) of the matrixcomplex 134c. Features of the frontal smiling overlay system 110c aredefined by fragments of this complex 220c comprising: (1) radialsconnecting V(D) to V(E), V(E) to V(J), V(J) to V(A), V(A) to V(F), V(F)to V(B), and V(B) to V(C); (2) a radial connecting V(B) to intersectV(C) R(288) I V(H) R(54); (3) a radial connecting V(E) to intersect V(D)R(72) I V(H) R(306); (4) a radial extending from V(B) at 72° andextending to intersect V(B) R(72) I V(G) R(18); (5) a radial extendingfrom V(E) at 288° and extending to intersect V(E) R(288) I V(I) R(342);(6) a radial extending from V(B) at 306° and extending to intersect V(B)R(306) I V(F) R(180); (7) a radial extending from V(E) at 54° andextending to intersect V(E) R(54) I V(J) R(180); and (8) a portion of aradial extending from V(B) at 270° and extending between intersect V(B)R(270) I V(F) R(162) and intersect V(B) R(270) I V(J) R(198).

2. Smiling External Dimple Complex

The smiling external dimple pentagon complex 230c (FIG. 21) ispositioned on the matrix pentagon complex 134c with V(H) of the dimplecomplex 230c coincident with V(H) of the matrix complex 134c and V(A)R(180) of the dimple complex 230c coincident with V(A) R(180) of thematrix complex 134c. Features of the frontal smiling overlay system 110care defined by fragments of this complex 230a comprising: (1) a radialextending from V(B) at 198° and extending to intersect dimple complexV(B) R(198) I matrix complex V(G) R(252); (2) a radial extending fromV(B) at 216° and extending to intersect V(B) R(216) I V(F) R(162); (3) aradial extending from V(E) at 162° and extending to intersect dimplecomplex V(E) R(162) I matrix complex V(I) R(108); (4) a radial extendingfrom V(E) at 144° and extending to intersect V(E) R(144) I V(J) R(198);(5) a portion of a radial extending from V(H) at 54° and extendingbetween intersect V(H) R(54) I V(C) R(18) and intersect V(H) R(54) IV(C) R(270); (6) a portion of a radial extending from V(C) at 270° andextending between intersect V(C) R(270) I V(H) R(54) and intersect V(C)R(270) I V(H) R(306); and (7) a portion of a radial extending from V(H)at 306° and extending between intersect V(H) R(306) I V(D) R(90) andintersect V(H) R(306) I V(D) R(342).

3. Smiling Dimple Complex

The smiling dimple pentagon complex 232c (FIG. 20) is positioned on thematrix pentagon complex 134c with V(H) of the dimple complex 232ccoincident with V(H) of the matrix complex 134c and V(A) R(180) of thedimple complex 232c coincident with V(A) R(180) of the matrix complex134c. Features of the frontal smiling overlay system 110c are defined byfragments of this complex 232c comprising: (1) a radial extending fromV(B) at 252° and extending to intersect V(B) R(252) I V(F) R(162); (2) aradial extending from V(E) at 108° and extending to intersect V(E)R(108) I V(J) R(198); (3) a radial extending from V(B) at 198° andextending to intersect V(B) R(198) I V(F) R(162); (4) a radial extendingfrom V(E) at 162° and extending to intersect V(E) R(162) I V(J) R(198);(5) a line connecting V(B) and intersect V(G) R(234) I V(C) R(324); (6)a line connecting V(E) and intersect V(I) R(126) I V(D) R(36); (7) aportion of a radial extending from V(B) at 234° and extending betweenintersect V(B) R(234) I V(C) R(324) and intersect dimple complex V(B)R(234) I matrix complex V(C) R(306); (8) a portion of a radial extendingfrom V(E) at 126° and extending between intersect V(E) R(126) I V(D)R(36) and intersect dimple complex V(E) R(126): I matrix complex V(D)R(54); and (9) a portion of a radial extending from V(B) at 270° andextending between intersect V(B) R(270) I V(A) R(144) and intersect V(B)R(270) I V(A) R(216).

4. Smiling Internal Smile Complex

The smiling internal smile pentagon complex 234c (FIG. 19) is positionedon the matrix pentagon complex 134c with V(A) R(180) of the smilecomplex 234c coincident to V(A) R(180) of the matrix complex 134c andmatrix complex 134c V(B) R(234) coincident to intersect smile complex234c V(A) R(126) I smile complex 234c V(E) R(72). Features of thefrontal smiling overlay system 110c are defined by fragments of thiscomplex 234c comprising: (1) a radial extending from V(B) at 288° andextending to intersect V(B) R(288) I V(F) R(198); (2) a radial extendingfrom V(E) at 72° and extending to intersect V(E) R(72) I V(J) R(162);(3) a radial extending from V(B) at 270° and extending to intersect V(B)R(270) I V(F) R(198); (4) a radial extending from V(E) at 90° andextending to intersect V(E) R(90) I V(J) R(162); (5) a radial extendingfrom V(B) at 234° and extending to intersect V(B) R(234) I V(A) R(162);(6) a radial extending from V(E) at 126° and extending to intersect V(E)R(126) I V(A) R(198); (7) a radial extending from V(B) at 216° andextending to intersect V(B) R(216) I matrix complex V(C) R(306); (8) aradial extending from V(E) at 144° and extending to intersect V(E)R(144) I matrix complex V(D) R(54); (9) a line extending betweenintersect V(B) R(234) I V(A) R(162) and intersect V(E) R(126) I V(A)R(198); (10) a point at intersect V(G) R(270) I matrix complex V(H)R(18); and (11) a point at intersect V(I) R(90) I matrix complex V(H)R(342).

IV. Lateral Smiling Overlay System

A lateral smiling overlay system 110d will be described in the samemanner as the overlay systems 110a, 110b, and 110c were described above.

A. Pentagon Complexes with n=0

The lateral smiling overlay system 110d includes only one pentagoncomplex of the size n=0. This pentagon complex has a Z value of 1 (seeFIG. 16) and is called a lateral smiling matrix pentagon complex 134d.

1. Lateral Smiling Matrix Complex

Features of the lateral smiling overlay system 110d are defined byfragments of the lateral smiling matrix pentagon complex 134dcomprising: (1) a radial extending from V(D) at 306° and extending tointersect V(D) R(306) I V(I) R(180); (2) a radial extending from V(I) at180° and extending to intersect V(I) R(180) I V(D) R(306); (3) a radialextending from V(E) at 162° and extending to intersect V(E) R(162) IV(D) R(108); (4) a portion of a radial extending from V(I) at 126° andextending between intersect V(I) R(126) I V(D) R(342) and intersect V(I)R(126) I V(D) R(54); (5) a portion of a radial extending from V(E) at144° and extending between intersect V(E) R(144) I V(I) R(18) andintersect V(E) R(144) I V(D) R(18); (6) a portion of a radial extendingfrom V(E) at 126° and extending between intersect V(E) R(126) I V(I)R(36) and intersect V(E) R(126) I V(D) R(18); (7) a line extending fromintersect V(E) R(126) I V(I) R(36) through intersect V(E) R(144) I V(I)R(18) and extending to the line described in (3) above; (8) a lineextending from intersect V(E) R(126) I V(I) R(36) to intersect V(E)R(90) I V(J) R(180); (9) a portion of a radial extending from V(E) at108° and extending between intersect V(E) R(108) I V(I) R(54) and theline described in (8) above; (10) a line extending between intersectV(J) R(180) I V(E) R(90) and intersect V(J) R(162) I V(A) R(216); (11) aportion of a radial extending from V(A) at 216° and extending betweenintersect V(A) R(216) I V(J) R(162) and intersect V(A) R(216) I V(J)R(144); (12) a line extending between intersect V(A) R(216) I V(J)R(144) and intersect V(A) R(234) I V(J) R(126); (13) a portion of aradial extending from V(A) at 234° and extending between intersect V(A)R(234) I V(J) R(126) and intersect V(A) R(234) I V(J) R(90); (14) apoint at intersect V(F) R(252) I V(A) R(180); (15) a radial extendingfrom V(D) at 288° and extending to intersect V(D) R(288) I V(E)R(166.5); (16) a line extending between intersect V(D) R(288) I V(E)R(166.5) and intersect V(D) R(270) I V(E) R(171); and (17) a lineextending between intersect V(D) R(270) I V(E) R(171) and intersect V(G)R(252) I V(E) R(180).

B. Pentagon Complexes with n=6

The overlay system 110d has one pentagon complex with n=6 (see FIG. 29).This pentagon complex 112 is a lateral smiling iris pentagon complex150d and is the same size as the iris pentagon complex 150 of thelateral repose overlay system 110b described above.

1. Smiling Iris Complex

The smiling iris pentagon complex 150d (FIG. 29) is positioned on thematrix complex 134d as is the iris pentagon complex 150 of the lateralrepose overlay system 110b described above. Features of the lateralsmiling overlay system 110c are defined by fragments of this irispentagon complex 150d which are the same as the fragments of the irispentagon complex 150 of the lateral repose overlay system 110b describedabove.

C. Pentagon Complexes with n=5

The lateral smiling overlay system 110d includes one pentagon complex ofthe size n=5. This pentagon complex is a lateral smiling internal narispentagon complex 160d and has a Z value of 1 (see FIG. 26).

1. Lateral Smiling Internal Naris Complex

The lateral smiling internal naris pentagon complex 160d is positionedon the matrix complex 134d as is the internal naris pentagon complex 160of the frontal repose overlay system 110a described above. Features ofthe lateral smiling overlay system 110d are defined by fragments of thispentagon complex 160d comprising: (1) a radial extending from V(F) at324° and extending to intersect V(F) R(324) I V(A) R(36); (2) a portionof a radial extending from V(J) at 180° and extending between intersectV(J) R(180) I V(E) R(54) and intersect V(J) R(180) I V(I) R(90); and (3)points at intersects: (a) V(B) R(0/360) I V(F) R(90); (b) V(A) R(216) IV(J) R(108); and (c) V(I) R(180) I V(D) R(288).

D. Pentagon Complexes with n=4

The lateral smiling overlay system 110d includes four pentagon complexesof the size n=4. These pentagon complexes are: (1) a lateral smilingnasal pentagon complex 170d; (2) a lateral smiling chin button pentagoncomplex 172d; (3) a lateral smiling eye pentagon complex 174d; and (4) alateral smiling teeth/lip eversion pentagon complex 224d. All four ofthese pentagon complexes have Z values of 1 (see FIG. 25).

1. Lateral Smiling Nasal Complex

The lateral smiling nasal pentagon complex 170d (FIG. 25) is positionedon the matrix pentagon complex 134d as is the nasal pentagon complex 170of the frontal repose overlay system 110a described above. Features ofthe lateral smiling overlay system 110d are defined by fragments of thispentagon complex 170d comprising: (1) a radial extending from V(I) at234° and extending to intersect V(I) R(234) I V(D) R(270); (2) a radialextending from V(I) at 126° and extending to intersect V(I) R(126) IV(D) R(342); (3) a radial extending from V(I) at 18° and extending tointersect V(I) R(18) I V(E) R(54); (4) a radial extending from V(G) at216° and extending to intersect V(G) R(216) I matrix complex V(G)R(270); (5) a portion of a radial extending from V(J) at 144° andextending between intersect V(J) R(144) I V(A) R(198) and intersect V(J)R(144) I V(G) R(252); (6) a portion of a radial extending from V(G) at270° and extending between intersect V(G) R(270) I V(F) R(198) andintersect V(G) R(270) I V(A) R(198); (7) a portion of a radial extendingfrom V(G) at 252° and extending between intersect V(G) R(252) I V(C)R(324) and intersect V(G) R(252) I V(C) R(306); (8) a portion of aradial extending from V(G) at 252° and extending between intersect V(G)R(252) I V(H) R(0/360) and intersect V(G) R(252) I V(H) R(306); (9) aline connecting intersect V(J) R(198) I V(E) R(72) and intersect V(J)R(162) I V(A) R(216); (10) a line connecting intersect V(J) R(162) IV(A) R(216) and intersect V(A) R(198) I V(B) R(270); (11) a portion of aradial extending from V(D) at 72° and extending between intersect V(D)R(72) I V(I) R(126) and intersect V(D) R(72) I V(I) R(108); (12) a lineextending between intersect V(I) R(108) I V(D) R(72) and intersect V(G)R(234) I V(C) R(324); (13) a line extending between intersect V(G)R(234) I V(C) R(324) and intersect nasal complex V(C) R(36) I matrixcomplex V(G) R(270); and (14) a radial extending from V(G) at 216° andextending to intersect V(G) R(216) I matrix complex V(G) R(270).

2. Lateral Smiling Chin Button Complex

The lateral smiling chin button pentagon complex 172d (FIG. 25) ispositioned on the matrix pentagon complex 134d as is the chin buttonpentagon complex 172 of the frontal repose overlay system 110a describedabove. Features of the lateral smiling overlay system 110d are definedby fragments of this pentagon complex 172d comprising: (1) a portion ofa radial extending from V(I) at 270° and extending between intersectV(I) R(270) I V(J) R(216) and intersect V(I) R(270) I V(J) R(234); (2) aportion of a radial extending from V(J) at 216° and extending betweenintersect V(J) R(216) I V(I) R(270) and intersect V(J) R(216) I V(I)R(324); (3) a portion of a radial extending from V(E) at 324° andextending between intersect V(E) R(324) I V(J) R(234) and intersect V(E)R(324) I V(A) R(270); and (4) a line connecting intersect V(E) R(324) IV(J) R(234) and intersect V(I) R(324) I V(E) R(216).

3. Lateral Smiling Eye Complex

The lateral smiling eye pentagon complex 174d (FIG. 25) is positioned onthe matrix pentagon complex 134d as is the eye pentagon complex 174 ofthe frontal repose overlay system 110a described above. Features of thelateral smiling overlay system 110d are defined by fragments of thispentagon complex 174d comprising: (1) a radial extending from V(E) at72° and extending to intersect V(E) R(72) I V(A) R(180); (2) a radialextending from V(E) at 216° and extending to intersect V(E) R(216) IV(I) R(324); (3) a radial extending from V(D) at 324° and extending tointersect V(D) R(324) I V(E) R(216); (4) a radial extending from V(D) at54° and extending to intersect V(D) R(54) I V(H) R(0/360); (5) a portionof a radial extending from V(B) at 270° and extending between intersectV(B) R(270) I V(A) R(162) and intersect V(B) R(270) I V(A) R(180); (6) aportion of a radial extending from V(H) at 324° and extending betweenintersect V(H) R(324) I V(D) R(54) and intersect V(H) R(324) I V(I)R(36); (7) a portion of a radial extending from matrix complex V(E) at108° and extending between intersect matrix complex V(E) R(108) I eyecomplex V(H) R(0/360) and intersect matrix complex V(E) R(108) I eyecomplex V(H) R(18); (8) a portion of a radial extending from V(E) at 90°and extending between intersect V(E) R(90) I V(J) R(162) and intersectV(E) R(90) I V(J) R(144); (9) a portion of a radial extending from V(J)at 144° and extending between intersect V(J) R(144) I V(F) R(234) andintersect V(J) R(144) I V(F) R(216); and (10) a portion of a radialextending from V(F) at 216° and extending between intersect V(F) R(216)I V(C) R(324) and intersect V(F) R(216) I V(C) R(306).

4. Lateral Smiling Teeth/Lip Eversion Complex

The lateral smiling teeth/lip eversion pentagon complex 224d (FIG. 25)is positioned on the matrix pentagon complex 134d as is the teeth/lipeversion pentagon complex 224b of the lateral repose overlay system 110bdescribed above. Features of the lateral smiling overlay system 110d aredefined by fragments of this pentagon complex 224d comprising: (1) aradial extending from V(E) at 0/360° and extending to intersect V(E)R(0/360) I V(J) R(234); (2) a portion of a radial extending from V(D) at270° and extending between intersect V(D) R(270) I V(I) R(216) andintersect V(D) R(270) I V(I) R(198); (3) a portion of a radial extendingfrom V(I) at 288° and extending between intersect V(I) R(288) I V(E)R(198) and intersect V(I) R(288) I V(E) R(252); (4) a portion of aradial extending from V(A) at 234° and extending between intersect V(A)R(234) I V(J) R(108) and intersect V(A) R(234) I V(J) R(198); and (5) apoint at intersect V(A) R(306) I V(J) R(18).

E. Pentagon Complexes with n=3

The lateral smiling overlay system 110d includes nine pentagon complexesof the size n=3. These pentagon complexes are: (1) a lateral smilingnose/mouth pentagon complex 184d; (2) a lateral smiling mouth/chinpentagon complex 186d; (3) a lateral smiling-chin inferior borderpentagon complex 188d; (4) a lateral smiling chin pentagon complex 190d;(5) a lateral smiling side chin pentagon complex 192d; (6) a lateralsmiling side nose/mouth pentagon complex 208d; (7) a lateral smilingeye/cheek pentagon complex 196d; (8) a lateral smiling eye brow pentagoncomplex 200d; and (9) a lateral smiling internal ear pentagon complex226d. All nine of these pentagon complexes have a Z value of 1 (see FIG.22).

1. Lateral Smiling Nose/Mouth Complex

The lateral smiling nose/mouth pentagon complex 184d (FIG. 22) ispositioned on the matrix pentagon complex 134d as is the nose/mouthpentagon complex 184 of the frontal repose overlay system 110a describedabove. Features of the lateral smiling overlay system 110d are definedby fragments of this pentagon complex 184d comprising: (1) a portion ofa radial extending from V(D) at 0/360° and extending between intersectV(D) R(0/360) I V(I) R(126) and intersect V(D) R(0/360) I V(I) R(108);(2) a radial extending from V(I) at 108° and extending to intersect V(I)R(108) I V(D) R(0/360); (3) a radial extending from V(I) at 288° andextending to intersect V(I) R(288) I V(E) R(198); (4) a radial extendingfrom V(I) at 18° and extending to intersect V(I) R(18) I V(E) R(162);(5) a portion of a radial extending from V(D) at 0/360° and extendingbetween intersect V(D) R(0/360) I V(I) R(90) and intersect V(D) R(0/360)I V(E) R(126); (6) a radial extending from V(E) at 90° and extending tointersect V(E) R(90) I V(A) R(198); (7) a portion of a radial extendingfrom V(I) at 54° and extending between intersect V(I) R(54) I V(J)R(162) and intersect V(I) R(54) I V(A) R(198); (8) a portion of a radialextending from V(A) at 198® and extending between intersect V(A) R(198)I V(I) R(54) and intersect V(A) R(198) I V(E) R(90); (9) a lineconnecting intersect V(A) R(198) I V(E) R(90) and intersect V(F) R(252)I V(J) R(126); (10) a portion of a radial extending from V(E) at 72° andextending between intersect V(E) R(72) I V(J) R(126) and intersect V(E)R(72) I V(J) R(108); (11) a line connecting intersect V(A) R(180) I V(J)R(108) and intersect V(A) R(144) I V(F) R(234); (12) a line connectingintersect V(A) R(144) I V(F) R(234) and intersect V(A) R(126) I V(F)R(216); (13) a radial extending from V(F) at 216° and extending tointersect V(F) R(216) I V(A) R(126); (14) a portion of a radialextending from V(F) at 306° and extending between intersect V(F) R(306)I V(J) R(0/360) and intersect V(F) R(306) I matrix complex V(F) R(234);(15) a radial extending from V(J) at 0/360° and extending to intersectV(J) R(0/360) I V(F) R(306); (16) a radial extending from V(J) at 18°and extending to intersect V(J) R(18) I V(A) R(306); (17) a radialextending from V(J) at 234° and extending to intersect V(J) R(234) IV(E) R(324); (18) a portion of a radial extending from matrix complexV(D) at 36° and extending between intersect matrix complex V(D) R(36) Inose/mouth complex V(E) R(270) and intersect matrix complex V(D) R(36) Inose/mouth complex V(E) R(324); (19) a line connecting intersect V(J)R(18) I V(A) R(306) and intersect V(J) R(0/360) I V(F) R(306); (20) aradial extending from V(J) at 126° and extending to intersect V(J)R(126) I V(A) R(234); (21) a portion of a radial extending from V(A) at234° and extending between intersect V(A) R(234) I V(J) R(126) andintersect V(A) R(234) I V(J) R(90); (22) a portion of a radial extendingfrom V(J) at 90° and extending between intersect V(J) R(90) I V(A)R(234) and intersect V(J) R(90) I V(A) R(144); (23) a portion of aradial extending from V(A) at 144° and extending between intersect V(A)R(144) I V(F) R(270) and intersect V(A) R(144) I V(F) R(252); (24) aportion of a radial extending from V(F) at 252° and extending betweenintersect V(F) R(252) I V(A) R(144) and intersect V(F) R(252) I V(A)R(162); (25) a radial extending from V(F) at 236° and extending tointersect V(F) R(236) I V(J) R(18); (26) a portion of a radial extendingfrom V(J) at 18° and extending between intersect V(J) R(18) I V(F)R(336) and intersect V(J) R(18) I matrix complex V(B) R(270); (27) aportion of a radial extending from V(F) at 351° and extending betweenintersect V(F) R(351) I matrix complex V(B) R(270) and intersect V(F)R(351) I matrix complex V(F) R(270); (28) a portion of a radialextending from matrix complex V(F) at 270° and extending betweenintersect matrix complex V(F) R(270) I nose/mouth complex V(F) R(351)and intersect matrix complex V(F) R(270) I matrix complex V(A) R(234).

Additional fragments may be described when this pentagon complex 184d isconsidered in conjunction with a lateral smiling smiling pentagoncomplex 236d (which is described below).

2. Lateral Smiling Mouth/Chin Complex

The lateral smiling mouth/chin pentagon complex 186d (FIG. 22) ispositioned on the matrix pentagon complex 134d as is the mouth/chinpentagon complex 186 of the frontal repose overlay system 110a describedabove. Features of the lateral smiling overlay system 110d are definedby fragments of this pentagon complex 186d comprising: (1) a radialextending from V(J) at 198° and extending to intersect V(J) R(198) IV(E) R(90); and (2) a portion of a radial extending from V(J) at 180°and extending between intersect V(J) R(180) I V(E) R(108) and intersectV(J) R(180) I V(E) R(126).

3. Lateral Smiling Chin Inferior Border Complex

The lateral smiling chin inferior border pentagon complex 188d (FIG. 22)is positioned on the matrix pentagon complex 134d as is the chininferior border pentagon complex 188 of the frontal repose overlaysystem 110a described above. Features of the lateral smiling overlaysystem 110d are defined by fragments of this pentagon complex 188dcomprising: (1) a radial extending from V(J) at 216° and extending tointersect V(J) R(216) I matrix complex V(C) R(288); (2) a portion of aradial extending from V(I) at 18° and extending between intersect V(I)R(18) I V(E) R(162) and intersect V(I) R(18) I V(E) R(90); and (3) aportion of a radial extending from V(D) at 270° and extending betweenintersect V(D) R(270) I V(I) R(198) and intersect V(D) R(270) I V(I)R(216).

4. Lateral Smiling Chin Complex

The lateral smiling chin pentagon complex 190d (FIG. 22) is positionedon the matrix pentagon complex 134d as is the chin pentagon complex 190of the frontal repose overlay system 110a described above. Features ofthe lateral smiling overlay system 110d are defined by fragments of thispentagon complex 190d comprising: (1) a radial extending from V(J) at180° and extending to intersect V(J) R(180) I V(A) R(234); (2) a portionof a radial extending from V(A) at 234° and extending between intersectV(A) R(234) I V(J) R(180) and intersect V(A) R(234) I V(J) R(198); and(3) a radial extending from V(E) at 144° and extending to intersect V(E)R(144) I V(I) R(54).

5. Lateral Smiling Side Chin Complex

The lateral smiling side chin pentagon complex 192d (FIG. 22) ispositioned on the matrix pentagon complex 134d as is the side chinpentagon complex 192 of the frontal repose overlay system 110a describedabove. Features of the lateral smiling overlay system 110d are definedby fragments of this pentagon complex 192d comprising: (1) a radialextending from V(I) at 1080 and extending to intersect V(I) R(108) IV(D) R(36); and (2) a portion of a radial extending from V(C) at 324°and extending between intersect V(C) R(324) I V(D) R(18) and intersectV(C) R(324) I matrix complex V(D) R(90).

6. Lateral Smiling Side Nose/Mouth Complex

The lateral smiling side nose/mouth pentagon complex 208d (FIG. 22) ispositioned on the matrix pentagon complex 134d as is the side nose/mouthpentagon complex 208 of the frontal repose overlay system 110a describedabove. Features of the lateral smiling overlay system 110d are definedby fragments of this pentagon complex 208d comprising: (1) a radialextending from V(H) at 342° and extending to intersect V(H) R(342) IV(I) R(90); (2) a portion of a radial extending from V(D) at 0/360° andextending between intersect V(D) R(0/360) I V(I) R(72) and intersectV(D) R(0/360) I V(I) R(36); (3) a portion of a radial extending fromV(I) at 36° and extending between intersect V(I) R(36) I V(E) R(108) andintersect V(I) R(36) I V(E) R(72); (4) a portion of a radial extendingfrom V(C) at 306° and extending between intersect V(C) R(306) I V(H)R(36) and intersect V(C) R(306) I V(D) R(36.); (5) a portion of a radialextending from V(D) at 36° and extending between intersect V(D) R(36) IV(C) R(306) and intersect V(D) R(36) I matrix complex V(C) R(288); and(6) a portion of a radial extending from matrix complex V(C) at 288° andextending between intersect matrix complex V(C) R(288) I nose/mouthcomplex V(D) R(36) and intersect matrix complex V(C) R(288) I nose/mouthcomplex V(H) R(0/360).

7. Lateral Smiling Eye/Cheek Complex

The lateral smiling eye/cheek pentagon complex 196d (FIG. 22) ispositioned on the matrix pentagon complex 134d as is the eye/cheekpentagon complex 196 of the frontal repose overlay system 110a describedabove. Features of the lateral smiling overlay system 110d are definedby fragments of this pentagon complex 196d comprising the same fragmentsof the eye/cheek complex 196b used to define features of the lateralrepose overlay system 110b described above.

8. Lateral Smiling Eye Brow Complex

The lateral smiling eye brow pentagon complex 200d (FIG. 22) ispositioned on the matrix pentagon complex 134d as is the eye browpentagon complex 200 of the frontal repose overlay system 110a describedabove. Features of the lateral smiling overlay system 110d are definedby fragments of this pentagon complex 200d comprising the same fragmentsof the eye brow complex 200b used to define features of the lateralrepose overlay system 110b described above.

9. Lateral Smiling Internal Ear Complex

The lateral smiling internal ear pentagon complex 226d is positioned onthe matrix pentagon complex 134d as is the internal ear pentagon complex134b of the lateral repose overlay system 110b described above. Featuresof the lateral smiling overlay system 110d are defined by fragments ofthis pentagon complex 226d comprising the same fragments of the internalear complex 226b used to define features of the lateral repose overlaysystem 110b described above.

F. Pentagon Complexes with n=2

The lateral smiling overlay system 110d includes five pentagon complexesof the size n=2. Three of these pentagon complexes have a Z value of 1(see FIG. 18): (1) a lateral smiling smile pentagon complex 220d; (2) alateral smiling smiling pentagon complex 236d; and (3) a lateral smilingear pentagon complex 228d. One has a Z value of (2/Φ)×(Φ^(1/3))! (seeFIG. 21): (4) a lateral smiling external dimple pentagon complex 230d.One has a Z value of 2/Φ (see FIG. 20): (5) a lateral smiling dimplepentagon complex 232d. The last one has a Z value of Φ/2 (see FIG. 19):(6) a lateral smiling internal smile pentagon complex 234d.

1. Lateral Smiling Smile Complex

The lateral smiling smile pentagon complex 220d (FIG. 18) is positionedon the matrix pentagon complex 134d as is the smile pentagon complex 220of the frontal repose overlay system 110a described above. Features ofthe lateral smiling overlay system 110d are defined by fragments of thispentagon complex 220d comprising: (1) a portion of a radial extendingfrom V(D) at 288° and extending between intersect V(D) R(288) I V(I)R(162) and intersect V(D) R(288) I V(I) R(252); and (2) a portion of aradial extending from V(E) at 126° and extending between intersect V(E)R(126) I V(I) R(18) and intersect V(E) R(126) I V(I) R(54).

2. Lateral Smiling Smiling Complex

The lateral smiling smiling pentagon complex 236d (FIG. 18) ispositioned on the matrix pentagon complex 134d with V(A) R(180) of thesmiling complex 236d coincident with V(A) R(180) of the matrix complex134d and V(F) R(234) of the smiling complex 236d coincident with V(B)R(234) of the matrix complex 134d. Features of the lateral smilingoverlay system 110d are defined by fragments of this pentagon complex236d comprising: (1) a radial extending from V(I) at 18° and extendingto intersect V(I) R(18) I V(E) R(90); (2) a portion of a radialextending from V(I) at 90° and extending between intersect V(I) R(90) IV(H) R(324) and intersect V(I) R(90) I V(D) R(18); (3) a portion of aradial extending from V(E) at 144° and extending between intersect V(E)R(144) I V(I) R(18) and intersect V(E) R(144) I V(I) R(90); and (4) aportion of a radial extending from matrix complex V(I) at 108° andextending between intersect matrix complex V(I) R(108) I smiling complexV(I) R(36) and intersect matrix complex V(I) R(108) I smiling complexV(I) R(72).

When considering this smiling complex 236d in conjunction with thelateral smiling nose/mouth pentagon complex 184d described above,additional fragments may be described comprising: (5) a radial extendingfrom nose/mouth complex V(E) at 252° and extending to intersectnose/mouth complex V(E) R(252) I smiling complex V(J) R(198); and (6) aline extending between intersect smiling complex V(A) R(198) I smilingcomplex V(F) R(234) and intersect smiling complex V(J) R(198) Inose/mouth complex V(E) R(252).

3. Lateral Smiling Ear Complex

The lateral smiling ear pentagon complex 228d (FIG. 18) is positioned onthe matrix pentagon complex 134d as is the ear pentagon complex 228b ofthe lateral repose overlay system 110b described above. Features of thelateral smiling overlay system 110d are defined by fragments of thispentagon complex 228d comprising the same fragments of the ear complex228b used to define features of the lateral repose overlay system 110bdescribed above.

4. Lateral Smiling External Dimple Complex

The lateral smiling external dimple pentagon complex 230d (FIG. 21) ispositioned on the matrix pentagon complex 134d as is the smilingexternal dimple pentagon complex 230c of the frontal smiling overlaysystem 110c described above. Features of the lateral smiling overlaysystem 110d are defined by fragments of this pentagon complex 230dcomprising: (1) a point at V(E). Additional fragments of this pentagoncomplex 230d may be defined if this complex 230d is taken intoconsideration in conjunction with a lateral smiling internal smilepentagon complex 234d described below.

5. Lateral Smiling Dimple Complex

The lateral smiling dimple pentagon complex 232d (FIG. 20) is positionedon the matrix pentagon complex 134d as is the smiling dimple pentagoncomplex 232c of the frontal smiling overlay system 110c described above.Features of the lateral smiling overlay system 110d are defined byfragments of this pentagon complex 232d comprising: (1) a point atintersect V(A) R(216) I V(E) R(126).

6. Lateral Smiling Internal Smile Complex

The lateral smiling internal smile pentagon complex 234d (FIG. 19) ispositioned on the matrix pentagon complex 134d as is the smilinginternal smile pentagon complex 234c of the frontal smiling overlaysystem 110c described above. Features of the lateral smiling overlaysystem 110d are defined by fragments of this pentagon complex 234dcomprising: (1) a radial extending from V(E) at 198° and extending tointersect V(E) R(198) I V(I) R(270).

When considering this complex 234d in conjunction with the lateralsmiling external dimple pentagon complex 230d described above,additional fragments may be described comprising: (2) a portion of aradial extending from dimple complex V(J) at 180° and extending betweenintersect dimple complex V(J) R(180) I dimple complex V(I) R(36) andintersect dimple complex V(J) R(180) I smiling complex V(E) R(198); and(3) a portion of a radial extending from V(I) at 36° and extendingbetween intersect dimple complex V(I) R(36) I dimple complex V(J) R(180)and intersect dimple complex V(I) R(36) I smiling complex V(E) R(198).

While an embodiment of the present invention has been shown anddescribed, various modifications may be made without departing from thescope of the present invention, and all such modifications andequivalents are intended to be covered.

I claim:
 1. A method to construct a three-dimensional geometric form todescribe the form of the human face comprisingproviding a first pentagoncomplex comprised of a pair of regular pentagons of equal size havingequal sides and equal internal angles and where said regular pentagonsare superimposed in an inverted relationship with a common center,creating radials comprised of lines drawn from each vertex of each ofsaid regular pentagons to each other vertex of each of said regularpentagons, forming at least a second pentagon complex wherein the sizeof said first and second pentagon complexes are related by the φproportion, and wherein said first and second pentagon complexes shareat least one common radial or intersect of said radials, selectingportions of said radials and intersects of said radials to create ageometric form which is comprised of major anthropometric points of thehuman face in a frontal view, selecting portions of said radials andintersects of said radials to create a geometric form which is comprisedof major anthropometric points of the human face in a lateral view, andcombining said frontal view geometric form with said lateral viewgeometric form to create a three-dimensional form describing the humanface.
 2. The method of claim 1 further comprising the step of analyzingthe face of a human by comparison with said three-dimensional geometricform.
 3. The method of claim 1 further comprising the step of alteringthe appearance of the human face to more closely coincide with saidthree-dimensional geometric form.
 4. The method of claim 1 wherein saidalteration is surgical.
 5. The method of claim 1 wherein said alterationis achieved by the application of cosmetics.
 6. A three-dimensionalgeometric overlay system to describe the form of an ideal human facecomprisinga first pentagon complex comprised of a pair of regularpentagons of equal size having equal sides and equal internal angles andwhere said regular pentagons are superimposed in an invertedrelationship with a common center, radials comprised of lines drawn fromeach vertex of each of said regular pentagons to each other vertex ofeach of said regular pentagons, at least a second pentagon complexwherein the size of said first and second pentagon complexes are relatedby the φ proportion, and wherein said first and second pentagoncomplexes share at least one common radial or intersect of said radials,portions of said radials and intersects of said radials create ageometric form comprised of major anthropometric points of a lateralview of the ideal human face, portions of said radials and intersects ofsaid radials create a geometric form comprised of major anthropometricpoints of a frontal view of the ideal human face, and means for applyingsaid frontal view geometric form and said lateral view geometric form tothe image of a subject's face for analyzing the subject's facialcomponents.
 7. The overlay system of claim 6 wherein said ideal humanface is a face in repose.
 8. The overlay system of claim 6 wherein saidideal human face is a smiling face.